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THE  ALGEBRA  OF  LOGIC 


BY 


LOUIS  COUTURAT 


AUTHORIZED  ENGLISH  TRANSLATION 


BY 


LYDIA  GILLINGHAM  ROBINSON,  B.  A. 
With  a  Preface  by  PHILIP  E.  B.  JOURDAIN.  M.  A.  (Cantab.) 


581— 


CHICAGO  AND  LONDON 

THE  OPEN  COURT  PUBLISHING  COMPANY 

1914 


Copyright  in  Great  Britain  under  the  Act  of  191 1, 


PREFACE. 

Mathematical  Logic  is  a  necessary  preliminary  to  logical 
Mathematics.  "Mathematical  Logic"  is  the  name  given  by 
Peano  to  what  is  also  known  (after  Venn)  as  "Symbolic 
Logic";  and  Symbolic  Logic  is,  in  essentials,  the  Logic  of 
Aristotle,  given  new  life  and  power  by  being  dressed  up  in 
the  wonderful — almost  magical — armour  and  accoutrements 
of  Algebra.  In  less  than  seventy  years,  logic,  to  use  an 
expression  of  De  Morgan's,  has  so  thriven  upon  symbols  and, 
in  consequence,  so  grown  and  altered  that  the  ancient  logicians 
would  not  recognize  it,  and  many  old-fashioned  logicians  will 
not  recognize  it.  The  metaphor  is  not  quite  correct:  Logic 
has  neither  grown  nor  altered,  but  we  now  see  more  of  it 
and  more  into  it. 

The  primary  significance  of  a  symbolic  calculus  seems  to 
lie  in  the  economy  of  mental  effort  which  it  brings  about,  and 
to  this  is  due  the  characteristic  power  and  rapid  development 
of  mathematical  knowledge.  Attempts  to  treat  the  operations 
of  formal  logic  in  an  analogous  way  had  been  made  not  in- 
frequently by  some  of  the  more  philosophical  mathematicians, 
such  as  Leibniz  and  Lambert;  but  their  labors  remained  little 
known,  and  it  was  Boole  and  De  Morgan,  about  the  middle 
of  the  nineteenth  century,  to  whom  a  mathematical — though 
of  course  non- quantitative — way  of  regarding  logic  was  due. 
By  this,  not  only  was  the  traditional  or  Aristotelian  doctrine 
of  logic  reformed  and  completed,  but  out  of  it  has  developed, 
in  course  of  time,  an  instrument  which  deals  in  a  sure  manner 
with  the  task  of  investigating  the  fundamental  concepts  of 
mathematics — a  task  which  philosophers  have  repeatedly  taken 
in  hand,  and  in  which  they  have  as  repeatedly  failed. 


IV  PREFACE. 

First  of  all,  it  is  necessary  to  glance  at  the  growth  of 
symbolism  in  mathematics^  where  alone  it  first  reached  per- 
fection. There  have  been  three  stages  in  the  development 
of  mathematical  doctrines:  first  came  propositions  with  par- 
ticular numbers,  like  the  one  expressed,  with  signs  subsequently 
invented,  by  "2  +  3  =  5";  then  came  more  general  laws  hold- 
ing for  all  numbers  and  expressed  by  letters,  such  as 

lastly  came  the  knowledge  of  more  general  laws  of  functions 
and  the  formation  of  the  conception  and  expression  "function". 
The  origin  of  the  symbols  for  particular  whole  numbers  is 
very  ancient,  while  the  symbols  now  in  use  for  the  operations 
and  relations  of  arithmetic  mostly  date  from  the  sixteenth  and 
seventeenth  centuries;  and  these  "constant"  symbols  together 
with  the  letters  first  used  systematically  by  Vi^te  (1540  — 1603) 
and  Descartes  (1596 — 1650),  serve,  by  themselves,  to  express 
many  propositions.  It  is  not,  then,  surprising  that  Descartes, 
who  was  both  a  mathematician  and  a  philosopher,  should 
have  had  the  idea  of  keeping  the  method  of  algebra  while 
going  beyond  the  material  of  traditional  mathematics  and 
embracing  the  general  science  of  what  thought  finds,  so  that 
philosophy  should  become  a  kind  of  Universal  Mathematics. 
This  sort  of  generalization  of  the  use  of  symbols  for  analogous 
theories  is  a  characteristic  of  mathematics,  and  seems  to  be 
a  reason  lying  deeper  than  the  erroneous  idea,  arising  from 
a  simple  confusion  of  thought,  that  algebraical  symbols  nec- 
essarily imply  something  quantitative,  for  the  antagonism 
there  used  to  be  and  is  on  the  part  of  those  logicians  who 
were  not  and  are  not  mathematicians,  to  symbolic  logic.  This 
idea  of  a  universal  mathematics  was  cultivated  especially  by 
Gottfried  Wilhelm  Leibniz  ( 1 646 —  1 7 1 6 ). 

Though  modem  logic  is  really  due  to  Boole  and  De 
Morgan,  Leibniz  was  the  first  to  have  a  really  distinct  plan 
of  a  system  of  mathematical  logic.  That  this  is  so  appears 
from  research — much  of  which  is  quite  recent — into  Leibniz's 
unpublished  worL 

The  principles  of  the   logic   of  Leibniz,   and  consequently 


PREFACE.  V 

of  his  whole  philosophy,  reduce  to  two':  (i)  All  our  ideas 
are  compounded  of  a  very  small  number  of  simple  ideas  which 
form  the  "alphabet  of  human  thoughts";  (2)  Complex  ideas 
proceed  from  these  simple  ideas  by  a  uniform  and  symmetrical 
combination  which  is  analogous  to  arithmetical  multiplication. 
With  regard  to  the  first  principle,  the  number  of  simple  ideas  is 
much  greater  than  Leibniz  thought;  and,  with  regard  to  the  second 
principle,  logic  considers  three  operations — which  we  shall  meet 
with  in  the  following  book  under  the  names  of  logical  multi- 
plication, logical  addition  and  negation — instead  of  only  one. 

"Characters"  were,  with  Leibniz,  any  written  signs,  and 
"real"  characters  were  those  which — as  in  the  Chinese  ideo- 
graphy — represent  ideas  directly,  and  not  the  words  for  them. 
Among  real  characters,  some  simply  serve  to  represent  ideas, 
and  some  serve  for  reasoning.  Egyptian  and  Chinese  hiero- 
glyphics and  the  symbols  of  astronomers  and  chemists  belong 
to  the  first  category,  but  Leibniz  declared  them  to  be  imper- 
fect, and  desired  the  second  category  of  characters  for  what 
he  called  his  "universal  characteristic".*  It  was  not  in  the 
form  of  an  algebra  that  Leibniz  first  conceived  his  charateristic, 
probably  because  he  was  then  a  novice  in  mathematics,  but 
in  the  form  of  a  universal  language  or  script.^  It  was  in 
1676  that  he  first  dreamed  of  a  kind  of  algebra  of  thought,'* 
and  it  was  the  algebraic  notation  which  then  served  as  model 
for  the  characteristics 

Leibniz  attached  so  much  importance  to  the  invention  of 
proper  symbols  that  he  attributed  to  this  alone  the  whole  of 
his  discoveries  in  mathematics.^  And,  in  fact,  his  infinitesimal 
calculus  affords  a  most  brilliant  example  of  the  importance 
of,  and  Leibniz's  skill  in  devising,  a  suitable  notation. ^ 

Now,  it  must  be  remembered  that  what  is  usually  understood 
by  the  name  "symbolic  logic",  and  which — though  not  its 
name — is  chiefly  due  to  Boole,  is  what  Leibniz  called  a 
Calculus   ratiocinator,    and   is    only    a   part  of  the  Universal 

^  CoUTURAT,  La  Logique  de  Leibniz  cFapres  des  documents  inidits, 
Paris,  1901,  pp.  431—432,  48. 

2  Ibid.,  p.  81.          3  Lbid.,  pp.   5 1,  78.  4  Ibid.,  p.  61. 

5  Ibid.,  p.  83.         6  Ibid.,  p.  84.  7  Ibid.,  p.  84—87. 


VI  PREFACE. 

Characteristic.  In  symbolic  logic  Leibniz  enunciated  the  principal 
properties  of  what  we  now  call  logical  multiplication,  addition, 
negation,  identity,  class-inclusion,  and  the  null-class;  but  the 
aim  of  Leibniz's  researches  was,  as  he  said,  to  create  "a  kind 
of  general  system  of  notation  in  which  all  the  truths  of  reason 
should  be  reduced  to  a  calculus.  This  could  be,  at  the  same 
time,  a  kind  of  universal  written  language,  very  different  from 
all  those  which  have  been  projected  hitherto;  for  the  char- 
acters and  even  the  words  would  direct  the  reason,  and  the 
errors — excepting  those  of  fact — would  only  be  errors  of 
calculation.  It  would  be  very  difficult  to  invent  this  language 
or  characteristic,  but  very  easy  to  learn  it  without  any 
dictionaries".  He  fixed  the  time  necessary  to  form  it:  "I  think 
that  some  chosen  men  could  finish  the  matter  within  five 
years";  and  finally  remarked:  "And  so  I  repeat,  what  I  have 
often  said,  that  a  man  who  is  neither  a  prophet  nor  a  prince 
can  never  undertake  any  thing  more  conducive  to  the  good 
of  the  human  race  and  the  glory  of  God". 

In  his  last  letters  he  remarked;  "If  I  had  been  less  busy, 
or  if  I  were  younger  or  helped  by  well-intentioned  young 
people,  I  would  have  hoped  to  have  evolved  a  characteristic 
of  this  kind";  and:  "I  have  spoken  of  my  general  characteristic 
to  the  Marquis  de  I'Hopital  and  others;  but  they  paid  no 
more  attention  than  if  I  had  been  telling  them  a  dream.  It 
would  be  necessary  to  support  it  by  some  obvious  use;  but, 
for  this  purpose,  it  would  be  necessary  to  construct  a  part 
at  least  of  my  characteristic; — and  this  is  not  easy,  above  all 
to  one  situated  as  I  am". 

Leibniz  thus  formed  projects  of  both  what  he  called  a 
characteristica  universalis,  aud  what  he  called  a  calculus  ratio- 
cinator;  it  is  not  hard  to  see  that  these  projects  are  inter- 
connected, since  a  perfect  universal  characteristic  would 
comprise,  it  seems,  a  logical  calculus.  Leibniz  did  not  pubUsh 
the  incomplete  results  which  he  had  obtained,  and  conse- 
quently his  ideas  had  no  continuators,  with  the  exception  of 
Lambert  and  some  others,  up  to  the  time  when  Boole,  De 
Morgan,  Schroder,  MacCoLL,  and  others  rediscovered  his 
theorems.     But  when   the  investigations   of  the  principles   of 


PREFACE.  VU 

mathematics  became  the  chief  task  of  logical  symbolism,  the 
aspect  of  symbolic  logic  as  a  calculus  ceased  to  be  of  such 
importance,  as  we  see  in  the  work  of  Frege  and  Russell, 
Frege's  symbolism,  though  far  better  for  logical  analysis  than 
Boole's  or  the  more  modern  Peano's,  for  instance,  is  far 
inferior  to  Peano's — a  symboUsm  in  which  the  merits  of 
internationality  and  power  of  expressing  mathematical  theorems 
are  very  satisfactorily  attained — in  practical  convenience. 
Russell,  especially  in  his  later  works,  has  used  the  ideas  of 
Frege,  many  of  which  he  discovered  subsequently  to,  but 
independently  of,  Frege,  and  modified  the  symbolism  of  Peano 
as  little  as  possible.  Still,  the  complications  thus  introduced 
take  away  that  simple  character  which  seems  necessary  to 
a  calculus,  and  which  Boole  and  others  reached  by  passing 
over  certain  distinctions  which  a  subtler  logic  has  shown  us 
must  ultimately  be  made. 

Let  us  dwell  a  little  longer  on  the  distinction  pointed  out 
by  Leibniz  between  a  calculus  ratiocinator  and  a  characteristica 
universalis  or  lingua  characteristica.  The  ambiguities  of  ordi- 
nary language  are  too  well  known  for  it  to  be  necessary  for 
us  to  give  instances.  The  objects  of  a  complete  logical 
symbolism  are:  firstly,  to  avoid  this  disadvantage  by  providing 
an  ideography,  in  which  the  signs  represent  ideas  and  the 
relations  between  them  directly  (without  the  intermediary  of 
words),  and  secondly,  so  to  manage  that,  from  given  premises, 
we  can,  in  this  ideography,  draw  all  the  logical  conclusions 
which  they  imply  by  means  of  rules  of  transformation  of 
formulas  analogous  to  those  of  algebra, — in  fact,  in  which 
we  can  replace  reasoning  by  the  almost  mechanical  process 
of  calculation.  This  second  requirement  is  the  requirement 
of  a  calculus  ratiocinator.  It  is  essential  that  the  ideo- 
graphy should  be  complete,  that  only  symbols  with  a  well- 
defined  meaning  should  be  used — to  avoid  the  same  sort  of 
ambiguities  that  words  have — and,  consequently,  that  no 
suppositions  should  be  introduced  implicitly,  as  is  commonly 
the  case  if  the  meaning  of  signs  is  not  well  defined.  Whatever 
premises  are  necessary  and  sufficient  for  a  conclusion  should 
be  stated  explicitly. 


Vm  PREFACE. 

Besides  this,  it  is  of  practical  importance, — though  it  is 
theoretically  irrelevant, — that  the  ideography  should  be  concise, 
so  that  it  is  a  sort  of  stenography. 

The  merits  of  such  an  ideography  are  obvious:  rigor  of 
reasoning  is  ensured  by  the  calculus  character;  we  are 
sure  of  not  introducing  unintentionally  any  premise;  and 
we  can  see  exactly  on  what  propositions  any  demonstration 
depends. 

We  can  shortly,  but  very  fairly  accurately,  characterize  the 
dual  development  of  the  theory  of  symbolic  logic  during  the 
last  sixty  years  as  follows:  The  calculus  ratiocinator  aspect 
of  symbolic  logic  was  developed  by  Boole,  De  Morgan, 
Jevons,  Venn,  C.  S.  Peirce,  Schroder,  Mrs.  Ladd  Franklin 
and  others;  the  lingua  characteristtca  aspect  was  developed 
by  Frege,  Peano  and  Russell.  Of  course  there  is  no  hard 
and  fast  boundary-line  between  the  domains  of  these  two 
parties.  Thus  Peirce  and  Schroder  early  began  to  work  at 
the  foundations  of  arithmetic  with  the  help  of  the  calculus  of 
relations;  and  thus  they  did  not  consider  the  logical  calculus 
merely  as  an  interesting  branch  of  algebra.  Then  Peano  paid 
particular  attention  to  the  calculative  aspect  of  his  symbolism. 
Frege  has  remarked  that  his  own  symbolism  is  meant  to  be 
a  calculus  ratiocinator  as  well  as  a  lingua  characteristica,  but 
the  using  of  Frege's  symbolism  as  a  calculus  would  be  rather 
like  using  a  three-legged  stand-camera  for  what  is  called 
"snap-shot"  photography,  and  one  of  the  outwardly  most 
noticeable  things  about  Russell's  work  is  his  combination  of 
the  symbolisms  of  Frege  and  Peano  in  such  a  way  as  to 
preserve  nearly  all  of  the  merits  of  each. 

The  present  work  is  concerned  with  the  calculus  ratiocinator 
aspect,  and  shows,  in  an  admirably  succinct  form,  the  beauty, 
symmetry  and  .simplicity  of  the  calculus  of  logic  regarded  as 
an  algebra.  In  fact,  it  can  hardly  be  doubted  that  some  such 
form  as  the  one  in  which  Schroder  left  it  is  by  far  the  best 
for  exhibiting  it  from  this  point  of  view.'    The  content  of  the 

1  Cf.  A.  N.  "Whitehead,  A  Treatise  en  Universal  Algebra  with  Appli- 
cations, Cambridge,  1898. 


PREFACE.  IX 

present  volume  corresponds  to  the  two  first  volumes  of 
Schroder's  great  but  rather  prolix  treatise.^  Principally  owing 
to  the  influence  of  C.  S.  Peirce,  Schroder  departed  from 
the  custom  of  Boole,  Jevons,  and  himself  (1877),  which 
consisted  in  the  making  fundamental  of  the  notion  of  equality, 
and  adopted  the  notion  of  subordination  or  inclusion  as  a 
primitive  notion.  A  more  orthodox  Boolian  exposition  is 
that  of  Venn',  which  also  contains  many  valuable  historical 
notes. 

We  will  finally  make  two  remarks. 

When  Boole  (cf.  S  2  below)  spoke  of  propositions  deter- 
mining a  class  of  moments  at  which  they  are  true,  he  really 
(as  did  MacColl)  used  the  word  "proposition"  for  what  we 
now  call  a  "propositional  function".  A  "proposition"  is  a 
thing  expressed  by  such  a  phrase  as  "twice  two  are  four"  or 
"twice  two  are  five",  and  is  always  true  or  always  false.  But 
we  might  seem  to  be  stating  a  proposition  when  we  say: 
"Mr.  William  Jennings  Bryan  is  Candidate  for  the  Presidency 
of  the  United  States",  a  statement  which  is  sometimes  true 
and  sometimes  false.  But  such  a  statement  is  like  a  mathe- 
matical function  in  so  far  as  it  depends  on  a  variable — the 
time.  Functions  of  this  kind  are  conveniently  distinguished 
from  such  entities  as  that  expressed  by  the  phrase  "twice 
two  are  four"  by  calling  the  latter  entities  "propositions"  and 
the  former  entities  "propositional  functions":  when  the  variable 
in  a  propositional  function  is  fixed,  the  function  becomes  a 
proposition.  There  is,  of  course,  no  sort  of  necessity  why 
these  special  names  should  be  used;  the  use  of  them  is 
merely  a  question  of  convenience  and  convention. 

In  the  second  place,  it  must  be  carefully  observed  that,  in 
8  13,  o  and  I  are  not  defined  by  expressions  whose  principal 

»  Vorlesungen  uber  die  Algebra  der  Logik,  Vol.  I.,  Leipsic,  1890; 
Vol.  II,  1 89 1  and  1905.  We  may  mention  that  a  much  shorter  Abriss 
of  the  work  has  been  prepared  by  EuGEN  MuLLER.  Vol.  Ill  (1895)  of 
Schr6der's  work  is  on  the  logic  of  relatives  founded  by  De  Morgan  and 
C.  S.  Peirce, — a  branch  of  Logic  that  is  only  mentioned  in  the  con- 
cluding sentences  of  this  volume. 

2  Symbolic  Logic,  London,   188 1;  2nd  ed.,  1894. 


X  PREFACE. 

copulas  are  relations  of  inclusion.  A  definition  is  simply  the 
convention  that,  for  the  sake  of  brevity  or  some  other  con- 
venience, a  certain  new  sign  is  to  be  used  instead  of  a  group 
of  signs  whose  meaning  is  already  known.  Thus,  it  is  the 
sign  of  equality  that  forms  the  principal  copula.  The  theory 
of  definition  has  been  most  minutely  studied,  in  modern  times 
by  Frege  and  Peano. 

Philip  E.  B.  Jourdain. 

Girton,  Cambridge.     England. 


CONTENTS. 

Page. 

Preface in 

Bibliography XIII 

1.  Introduction     3 

2.  The  Two  Interpretations  of  the  Logical  Calculus 3 

3.  Relation  of  Inclusion        4 

4.  Definition  of  Equality      6 

5.  Principle  of  Identity         8 

6.  Principle  of  the  Syllogism       8 

7.  Multiplication  and  Addition     9 

8.  Principles  of  Simplification  and  Composition 11 

9.  The  Laws  of  Tautology  and  of  Absorption 12 

10.  Theorems  on  Multiplication  and  Addition        14 

11.  The  First  Formula  for  Transforming  Inclusions  into  Equalities  15 

12.  The  Distributive  Law 16 

13.  Definition  of  o  and  I      17 

14.  The  Law  of  Duality        ... 20 

15.  Definition  of  Negation     21 

16.  The  Principles  of  Contradiction  and  of  Excluded  Middle         ...  23 

17.  Law  of  Double  Negation         24 

18.  Second    Formula    for    Transforming  Inclusions  into    Equalities  25 

19.  Law  of  Contraposition 26 

20.  Postulate  of  Existence     ,..  27 

21.  The  Developments  of  o  and  of  i 28 

22.  Properties  of  the  Constituents          29 

23.  Logical  Ftmctions 29 

24.  The  Law  of  Development       30 

25.  The  Formulas  of  De  Morgan            32 

26.  Disjunctive  Sums     34 

27.  Properties  of  Developed  Functions           34 

28.  The  Limits  of  a  Function       ,.  37 

29.  Formula  of  Poretsky        38 

30.  Schroder's  Theorem         39 

31.  The  Resultant  of  Elimination           41 

32.  The  Case  of  Indetermination 43 


XII  CONTENTS. 

Page. 

32.  Sums  and  Products  of  Functions     44 

34.  The  Expression  of  an  Inclusion  by  Means  of  an  Indeterminate  46 

35.  The  Expression   of  a  Double  Inclusion  by  Means  of  an  Inde- 

terminate        48 

36.  Solution  of  an  Equation  Involving  One  Unknown  Quantity     ...  50 

37.  Elimination  of  Several  Unknown  Quantities     53 

38.  Theorem  concerning  the  Values  of  a  Function        ...       55 

39.  Conditions  of  Impossibility  and  Indetermination      57 

40.  Solution  of  Equations  Containing  Several  Unknown  Quantities  57 

41.  The  Problem  of  Boole 59 

42.  The  Method  of  Poretsky          61 

43.  The  Law  of  Forms         62 

44.  The  Law  of  Consequences      63 

45.  The  Law  of  Causes         67 

46.  Forms  of  Consequences  and  Causes        69 

47.  Example:  Venn's  Problem       70 

48.  The  Geometrical  Diagrams  of  Venn        73 

49.  The  Logical  Machine  of  Jevons     75 

50.  Table  of  Consequences 76 

51.  Table  of  Causes      •       77 

52.  The  Number  of  Possible  Assertions         79 

53.  Particular  Propositions     * 80 

54.  Solution  of  an  Inequation  with  One  Unknown           „       „       ...  81 
55-  System  of  an  Equation  and  an  Inequation      83 

56.  Formulas  Peculiar  to  the  Calculus  of  Propositions 84 

57.  Equivalence  of  an  Implication  and  an  Alternative 85 

58.  Law  of  Importation  and  Exportation      88 

59.  Reduction  of  Inequalities  to  Equalities    90 

60.  Conclusion      92 

Index 95 


BIBLIOGRAPHY/ 

George  Boole.  The  Mathematical  Analysis  of  Logic  (Cam- 
bridge and  London,  1847). 

—  An  Investigation  of  the  Laws  of  Thought  (London  and 
Cambridge,  1854). 

W,  Stanley  Jevons,     Fure  Logic  (London,   1864). 

—  "On  the  Mechanical  Performance  of  Logical  Inference" 
{Philosophical  Transactions^  1870), 

Ernst  Schroder.  Der  Oj>erationskreis  des  Logikkalkuls 
(Leipsic,  1877). 

—  Vorlesungen  iiber  die  Algebra  der  Logik,  Vol.  I  (1890), 
VoL  n  (1891),  Vol.  Ill:  Algebra  und  Logik  der  Relative 
(1895)  (Leipsic).^ 

Alexander  Macfarlane.  Principles  of  the  Algebra  of  Logic, 
with  Examples  (Edinburgh,   1879). 

John  Venn.  Symbolic  Logic,  1881;  2nd.  ed.,  1894  (London).^ 
Studies  in  Logic  by  members  of  the  Johns  Hopkins  Uni- 
versity (Boston,  1883):  particularly  Mrs.  Ladd-Franklin, 
O.  Mitchell  and  C.  S.  Peirce. 

A.  N.  Whitehead.  A  Treatise  on  Universal  Algebra.  Vol.  I 
(Cambridge,  1898). 

—  "Memoir  on  the  Algebra  of  Symbolic  Logic"  {American 
Journal  of  Mathematics,  Vol.  XXIU,  1901). 


1  This  list  contains  only  the  works  relating  to  the  system  of  Boole 
and  Schroder  explained  in  this  work. 

2  EuGEN  MuLLER  has  prepared  a  part,  and  is  preparing  more,  of 
the  publication  of  supplements  to  Vols.  11  and  III,  from  the  papers  left 
by  Schr6der. 

3  A  valuable  work  from  the  points  of  view  of  history  and  bibliog- 
raphy. 


XIV  BIBLIOGRAPHY. 

EuGEN  MiJLLER.  Ober  die  Algebra  der  Logik:  I.  Die  Grund- 
lagen  des  Gebietekalkuls ;  II.  Das  Elimitiationsproblem  und 
die  Syllogistik;  Programs  of  the  Grand  Ducal  Gymnasium 
of  Tauberbischofsheim  (Baden),   1900,   1901   (Leipsic), 

W.  E.  Johnson.  "Sur  la  theorie  des  egalites  logiques"  {Biblio- 
tKeqiie  du  Congres  international  de  Philosophie.  Vol.  Ill, 
Logique  et  Histoire  des  Sciences;  Paris,   1901). 

Platon  Poretsky.  Sept  Lois  fondamentales  de  la  theorie  des 
Sgalitis  logiques  (Kazan,   1899). 

—  Quelques  lois  ult^ieures  de  la  theorie  des  egalith  logiques 
(Kazan,  1902). 

—  ''Expos^  elementaire  de  la  theorie  des  egalites  logiques  a 
deux  termes"  {Revue  de  Metaphysique  et  de  Morale.  Vol.  VIII, 
1900.) 

—  "Theorie  des  egalites  logiques  a  trois  termes"  {Bibliotheqiie 
du  Congres  international  de  Philosophie.  Vol.  III.  {Logique 
et  Histoire  des  Sciences.     (Paris,   1901,  pp.   201 — 233). 

—  Theorie  des  non-Sgalitds  logiques  (Kazan,   1904). 

E.  V.  Huntington.  "Sets  of  Independent  Postulates  for  the 
Algebra  of  Logic"  {Transactions  of  the  American  Mathe- 
matical Society,  Vol.  V,  1904). 


THE  ALGEBRA  OF  LOGIC. 


1.  Introduction. — The  algebra  of  logic  was  founded  by 
George  Boole  (1815 — 1864);  it  was  developed  and  perfected 
by  Ernst  Schroder  (1841  — 1902).  The  fundamental  laws 
of  this  calculus  were  devised  to  express  the  principles  of 
reasoning,  the  "laws  of  thought".  But  this  calculus  may  be 
considered  from  the  purely  formal  point  of  view,  which  is 
that  of  mathematics,  as  an  algebra  based  upon  certain  prin- 
ciples arbitrarily  laid  down.  It  belongs  to  the  realm  of 
philosophy  to  decide  whether,  and  in  what  measure,  this 
calculus  corresponds  to  the  actual  operations  of  the  mind, 
and  is  adapted  to  translate  or  even  to  replace  argument; 
we  cannot  discuss  this  point  here.  The  formal  value  of  this 
calculus  and  its  interest  for  the  mathematician  are  absolutely 
independent  of  the  interpretation  given  it  and  of  the  appli- 
cation which  can  be  made  of  it  to  logical  problems.  In 
short,  we  shall  discuss  it  not  as  logic  but  as  algebra. 

2.  The  Two  Interpretations  of  the  Logical  Cal- 
culus.— There  is  one  circumstance  of  particular  interest, 
namely,  that  the  algebra  in  question,  like  logic,  is  susceptible 
of  two  distinct  interpretations,  the  parallelism  between  them 
being  almost  perfect,  according  as  the  letters  represent  con- 
cepts or  propositions.  Doubtless  we  can,  with  Boole  and 
Schroder,  reduce  the  two  interpretations  to  one,  by  con- 
sidering the  concepts  on  the  one  hand  and  the  propositions 
on  the  other  as  corresponding  to  assemblages  or  classes;  since 
a  concept  determines  the  class  of  objects  to  which  it  is 
applied  (and  which  in  logic  is  called  its  extension),  and  a 
proposition  determines  the  class  of  the  instances  or  moments 
of  time  in  which  it  is  true  (and  which  by  analogy  can  also 
be    called    its    extension).     Accordingly   the   calculus  of  con- 

I* 


4  LOGICAL    CALCULUS    AND    INCLUSION. 

cepts  and  the  calculus  of  propositions  become  reduced  to 
but  one,  the  calculus  of  classes,  or,  as  Leibniz  called  it,  the 
theory  of  the  whole  and  part,  of  that  which  contains  and 
that  which  is  contained.  But  as  a  matter  of  fact,  the  cal- 
culus of  concepts  and  the  calculus  of  propositions  present 
certain  differences,  as  we  shall  see,  which  prevent  their  com- 
plete identification  from  the  formal  point  of  view  and  conse- 
quently their  reduction  to  a  single  "calculus  of  classes". 

Accordingly  we  have  in  reality  three  distinct  calculi,  or, 
in  the  part  common  to  all,  three  different  interpretations  of 
the  same  calculus.  In  any  case  the  reader  must  not  forget 
that  the  logical  value  and  the  deductive  sequence  of  the 
formulas  does  not  in  the  least  depend  upon  the  inter- 
pretations which  may  be  given  them,  and,  in  order  to 
make  this  necessary  abstraction  easier,  we  shall  take  care  to 
place  the  symbols  "C.  I."  {conceptual  interpretation)  and  "P.  I." 
{prepositional  interpretation)  before  all  interpretative  phrases. 
These  interpretations  shall  serve  only  to  render  the  formulas 
intelligible,  to  give  them  clearness  and  to  make  their  mean- 
ing at  once  obvious,  but  never  to  justify  them.  They  may 
be  omitted  without  destroying  the  logical  rigidity  of  the 
system. 

In  order  not  to  favor  either  interpretation  we  shall  say 
that  the  letters  represent  terms;  these  terms  may  be  either 
concepts  or  propositions  according  to  the  case  in  hand. 
Hence  we  use  the  word  term  only  in  the  logical  sense. 
When  we  wish  to  designate  the  "terras"  of  a  sum  we  shall 
use  the  word  summand  in  order  that  the  logical  and  mathe- 
matical meanings  of  the  word  may  not  be  confused.  A  term 
may  therefore  be  either  a  factor  or  a  summand. 

3.  Relation    of  Inclusion.— Like   all   deductive   theories, 

.the   algebra   of  logic   may  be  established  on  various  systems 

of  principles';    we   shall,  choose   rhe  one  which  most  nearly 

»  See  Huntington,  "Sets  of  Independent  Postulates  for  the  Algebra 
of  Logic",  Transaclions  0/  the  Am.  Math.  Soc,  Vol.  V,  1904,  pp.  288 — 309. 
[Here  he  says:  "Any  set  o£  consistent  postulates  would  give  rise  to  a 
corresponding   algebra,     viz.,    the   totality    of  propositions   which    follow 


RELATION    OF    INCLUSION.  5 

approaches  the  exposition  of  Schroder  and  current  logical 
interpretation. 

The  fundamental  relation  of  this  calculus  is  the  binary 
(two -termed)  relation  which  is  called  inclusion  (for  classes), 
subsumption  (for  concepts),  or  implication  (for  propositions). 
We  will  adopt  the  first  name  as  affecting  alike  the  two  logical 
interpretations,  and  we  will  represent  this  relation  by  the 
sign  <C[  because  it  has  formal  properties  analogous  to  those 
of  the  mathematical  relation  <^  ("less  than")  or  more  exactly 
< ,  especially  the  relation  of  not  being  symmetrical.  Because 
of  this  analogy  Schroder  represents  this  relation  by  the  sign  =^ 
which  we  shall  not  employ  because  it  is  complex,  whereas 
the  relation  of  inclusion  is  a  simple  one. 

In  the  system  of  principles  which  we  shall  adopt,  ^this 
relation  is  taken  as  a  primitive  idea  and  is  consequently 
indefinable.  The  explanations  which  follow  are  not  given 
for  the  purpose  of  definifig  it  but  only  to  indicate  its  meaning 
according  to  each  of  the  two  interpretations. 

C.  I.:  When  a  and  b  denote  concepts,  the  relation  a  <C  ^ 
signifies  that  the  concept  a  is  subsumed  under  the  concept  b; 
that  is,  it  is  a  species  with  respect  to  the  genus  b.  From 
the  extensive  point  of  view,  it  denotes  that  the  class  of  a'% 
is  contained  in  the  class  of  b'%  or  makes  a  part  of  it;  or, 
more  concisely,  that  "All  a's  are  ^'s".  From  the  comprehen- 
sive point  of  view  it  means  that  the  concept  b  is  contained 
in  the  concept  a  or  makes  a  part  of  it,  so  that  consequently 
the  character  a  implies  or  involves  the  character  b.  Example: 
"All  men  are  mortal";  "Man  imphes  mortal";  "Who  says 
man  says  mortal";  or,  simply,  "Man,  therefore  mortal". 

P.  I. :  When  a  and  b  denote  propositions,  the  relation  a  <^b 
signifies  that  the  proposition  a  implies  or  involves  the  prop- 
osition b,  which  is  often  expressed  by  the  hypothetical 
judgment,  "If  a  is  true,  b  is  true";  or  by  "a:  implies  b";  or 
more  simply  by  "«,  therefore  b".     We  see  that  in  both  inter- 


from  these  postulates  by  logical  deductions.  Every  set  of  postulates  should 
be  free  from  redundances,  in  other  words,  the  postulates  of  each  set 
should  be  independetti,  no  one  of  them  deducible  from  the  rest."] 


6  DEFINITION    OF    EQUALITY. 

pretations  the  relation  <^  may  be  translated  approximately 
by  "therefore". 

Remark. — Such  a  relation  as  "<3;  <^  V  is  a  proposition, 
whatever  may  be  the  interpretation  of  the  terms  a  and  b. 
Consequently,  whenever  a  <^  relation  has  two  like  relations 
(or  even  only  one)  for  its  members,  it  can  receive  only  the 
propositional  interpretation,  that  is  to  say,  it  can  only  denote 
an  implication. 

A  relation  whose  members  are  simple  terms  (letters)  is 
called  a  primary  proposition;  a  relation  whose  members  are 
primary  propositions  is  called  a  secondary  proposition,  and 
so  on. 

From  this  it  may  be  seen  at  once  that  the  propositional 
interpretation  is  more  homogeneous  than  the  conceptual, 
since  it  alone  makes  it  possible  to  give  the  same  meaning 
to  the  copula  <<]  in  both  primary  and  secondary  prop- 
ositions. 

4.  Definition  of  Equality. — There  is  a  second  copula 
that  may  be  defined  by  means  of  the  first;  this  is  the 
copula  =   ("equal  to").     By  definition  we  have 

a  ==  b, 
whenever 

a  <^b  and  b  <^a 

are  true  at  the  same  time,  and  then  only.  In  other  words, 
the  single  relation  a  =  ^  is  equivalent  to  the  two  simulta- 
neous relations  a  <ib  and  b  <Ca. 

In  both  interpretations  the  meaning  of  the  copula  =  is 
determined  by  its  formal  definition: 

C.  I:  a  =  b  means,  "All  a's  are  ^'s  and  all  b's  are  <z's"; 
in  other  words,  that  the  classes  a  and  b  coincide,  that  they 
are  identical.* 

P.  I.:  a  =  b  means   that   a   implies  b  and  b  implies  a;    in 

I  This  does  not  mean  that  the  concepts  a  and  b  have  the  same 
meaning.  Examples:  "triangle"  and  "trilateral",  "equiangular  triangle" 
and  "equilateral  triangle". 


DEFINITION    OF    EQUALITY,  7 

Other  words,  that  the  propositions  a  and  b  are  equivalent, 
that  is  to  say,  either  true  or  false  at  the  same  time.* 

Remark.— Tht  relation  of  equality  is  symmetrical  by  very 
reason  of  its  definition:  a  =  ^  is  equivalent  to  b  =  a.  But 
the  relation  of  inclusion  is  not  symmetrical:  a<ib  \s  not 
equivalent  to  b<^a,  nor  does  it  imply  it.  We  might  agree 
to  consider  the  expression  a'^  b  equivalent  to  b  <Ca,  but 
we  prefer  for  the  sake  of  clearness  to  preserve  always  the 
same  sense  for  the  copula  <^.  However,  we  might  translate 
verbally  the  same  inclusion  a<Cb  sometimes  by  "a  is  con- 
tained in  V  and  sometimes  by  "^  contains  a". 

In  order  not  to  favor  either  interpretation,  we  will  call 
the  first  member  of  this  relation  the  antecedent  and  the 
second  the  consequent. 

C.  L:  The  antecedent  is  the  subject  and  the  consequent  is 
the  predicate  of  a  universal  affirmative  proposition. 

P.  L :  The  antecedent  is  the  premise  or  the  cause,  and  the 
consequent  is  the  consequence.  When  an  implication  is  trans- 
lated by  a  hypothetical  (or  conditional)  judgment  the  ante- 
cedent is  called  the  hypothesis  (or  the  condition)  and  the 
consequent  is  called  the  thesis. 

When  we  shall  have  to  demonstrate  an  equality  we  shall 
usually  analyze  it  into  two  converse  inclusions  and  demon- 
strate them  separately.  This  analysis  is  sometimes  made  also 
when  the  equality  is  a  datum  (a  premise). 

When  both  members  of  the  equality  are  propositions,  it 
can  be  separated  into  two  impUcations,  of  which  one  is 
called  a  theorem  and  the  other  its  reciprocal.  Thus  when- 
ever a  theorem  and  its  reciprocal  are  true  we  have  an 
equality.  A  simple  theorem  gives  rise  to  an  implication 
whose  antecedent  is  the  hypothesis  and  whose  consequent  is 
the  thesis  of  the  theorem. 

It  is  often  said  that  the  hypothesis  is  the  sufficient  condition 
of  the  thesis,  and  the  thesis  the  necessary  condition  of  the  hy- 


I  This  does  not  mean  that  they  have  the  same  meaning.  Example: 
"The  triangle  ABC  has  two  equal  sides",  and  "The  triangle  ABC  has 
two  equal  angles". 


8  IDENTITY   AND    SYLLOGISM. 

pothesis;  that  is  to  say,  it  is  sufficient  that  the  hypothesis  be 
true  for  the  thesis  to  be  true;  while  it  is  necessary  that  the 
thesis  be  true  for  the  hypothesis  to  be  true  also.  When  a 
theorem  and  its  reciprocal  are  true  we  say  that  its  hypoth- 
esis is  the  necessary  and  sufficient  condition  of  the  thesis; 
that  is  to  say,  that  it  is  at  the  same  time  both  cause  and 
consequence. 

5.  Principle  of  Identity. — The  first  principle  or  axiom 
of  the  algebra  of  logic  is  the  principle  of  identity^  which  is 
formulated  thus: 

(Ax.  I)  a<Ca^ 

whatever  the  term  a  may  be. 

C.  L:  "All  a's  are  a's",  i.  e.,  any  class  whatsoever  is  con- 
tained in  itself 

P.  I.:  "a  implies  a",  i.  e.,  any  proposition  whatsoever  im- 
plies itself 

This  is  the  primitive  formula  of  the  principle  of  identity. 
By  means  of  the  definition  of  equality,  we  may  deduce  from 
it  another  formula  which  is  often  wrongly  taken  as  the  ex- 
pression of  this  principle: 

a  =  a, 
whatever  a  may  be;  for  when  we  have 

we  have  as  a  direct  result,  t 

a  =  a. 
C.  I.:  The  class  a  is  identical  with  itself 
P.  I.:  The  proposition  a  is  equivalent  to  itself 

6.  Principle  of  the  Syllogism. — Another  principle  of 
the  algebra  of  logic  is  the  principle  of  the  syllogism^  which 
may  be  formulated  as  follows: 

(Ax.  n)  {a<,b)     (^<^)<(^<4 

C.  I.:  "If  all  a's  are  ^'s,  and  if  all  ^'s  are  ^'s,  then  all  a's 
are  ^'s".     This  is  the  principle  of  the  categorical  syllogism. 

P.  L:  "If  a  implies  b^  and  if  b  implies  c,  a  implies  <r." 
This  is  the  principle  of  the  hypothetical  syllogism. 


MULTIPLICATION    AND    ADDITION.  9 

We  see  that  in  this  formula  the  principal  copula  has  al- 
ways the  sense  of  implication  because  the  proposition  is  a 
secondary  one. 

By  the  definition  of  equality  the  consequences  of  the 
principle  of  the  syllogism  may  be  stated  in  the  following 
formulas ': 

{a<b)     (b=.c)<(a<c), 

(a  =  d)     {b<c)<^{a<c\ 
(a  =  b)     {b  =  c)<,{a  =  c). 

The  conclusion  is  an  equality  only  when  both  premises 
are  equalities. 

The  preceding  formulas  can  be  generalized  as  follows: 
{a<,b)     ib<c)     (c<d)<(a<:d), 
{a  =  b)     {b^  c)     {c=^  dX_{a  =  d). 

Here  we  have  the  two  chief  formulas  of  the  sorites.  Many 
other  combinations  may  be  easily  imagined,  but  we  can  have 
an  equality  for  a  conclusion  only  when  all  the  premises  are 
equalities.  This  statement  is  of  great  practical  value.  In  a 
succession  of  deductions  we  must  pay  close  attention  to  see 
if  the  transition  from  one  proposition  to  the  other  takes  place 
by  means  of  an  equivalence  or  only  of  an  implication.  There 
is  no  equivalence  between  two  extreme  propositions  unless 
all  intermediate  deductions  are  equivalences;  in  other  words, 
if  there  is  one  single  implication  in  the  chain,  the  relation 
of  the   two    extreme   propositions  is  only  that  of  implication. 

7.  Multiplication  and  Addition. — The  algebra  of  logic 
admits  of  three  operations,  logical  multiplication,  logical  addition, 
and  negation.  The  two  former  are  binary  operations,  that  is 
to  say,  combinations  of  two  terms  having  as  a  consequent  a 
third  term  which  may  or  may  not  be  different  from  each  of 
them.  The  existence  of  the  logical  product  and  logical  sum 
of   two    terms    must    necessarily    answer    the    purpose    of   a 

X  Strictly  speaking,  these  formulas  presuppose  the  laws  of  multi- 
plication which  will  be  established  further  on;  but  it  is  fitting  to  cite 
them  here  in  order  to  compare  them  with  the  principle  of  the  syllogism 
from  which  they  are  derived. 


lO  MULTIPLICATION    AND    ADDITION. 

double  postulate,  for  simply  to  define  an  entity  is  not  enough 
for  it  to  exist.    The  two  postulates  may  be  formulated  thus: 
(Ax.  in).    Given    any  two  terms,  a  and  b,  then  there  is  a 
term  /  such  that 

and  that  for  every  value  of  x  for  which 

x<^a,  x<.b, 
we  have  also 

x<p. 

(Ax.  IV).  Given  any  two  terms,  a  and  b,  then  there  exists 
a  term  s  such  that 

a<j-,  b<:,s, 

and  that,  for  any  value  of  x  for  which 

a<^x,  b  <Cx, 
we  have  also 

s  <^x. 

It  is  easily  proved  that  the  terms  /  and  s  determined  by 
the  given  conditions  are  unique,  and  accordingly  we  can 
define  the  product  ab  and  the  sum  a  +  ^  as  being  respec- 
tively the  terms  /  and  s. 

C.  I.:  I.  The  product  of  two  classes  is  a  class  /  which 
is  contained  in  each  of  them  and  which  contains  every 
(other)  class  contained  in  each  of  them; 

2.  The  sum  of  two  classes  a  and  ^  is  a  class  s  which 
contains  each  of  them  and  which  is  contained  in  every  (other) 
class  which  contains  each  of  them. 

Taking  the  words  "less  than"  and  "greater  than"  in  a  meta- 
phorical sense  which  the  analogy  of  the  relation  <C  with  the 
mathematical  relation  of  inequality  suggests,  it  may  be  said 
that  the  product  of  two  classes  is  the  greatest  class  contained 
in  both,  and  the  sum  of  two  classes  is  the  smallest  class 
which    contains    both.'      Consequently    the    product    of   two 

I  According  to  another  analogy  Dedekind  designated  the  logical  sum 
and  product  by  the  same  signs  as  the  least  common  multiple  and  greatest 
common  divisor  {Was  sind  und  was  sollen  die  Zahlen?  Nos.  8  and  17,  1887. 
[Cf.  English  translation  entitled  Essays  on  Number  (Chicago,  Open  Court 
Publishing  Co.  1901,  pp.  46  and  48)  ]  Georg  Cantor  originally  gave 
them  the  same  designation  {Mathematische  Annalen,  Vol.  XVII,  1880). 


SIMPLIFICATION    AND    COMPOSITION.  II 

classes  is  the  part  that  is  common  to  each  (the  class  of 
their  common  elements)  and  the  sum  of  two  classes  is  the 
class  of  all  the  elements  which  belong  to  at  least  one 
of  them. 

P.  L:  I.  The  product  of  two  propositions  is  a  proposition 
which  implies  each  of  them  and  which  is  implied  by  every 
proposition  which  imphes  both: 

2.  The  sum  of  two  propositions  is  the  proposition  which 
is  implied  by  each  of  them  and  which  implies  every  prop- 
osition implied  by  both. 

Therefore  we  can  say  that  the  product  of  two  propositions 
is  their  weakest  common  cause,  and  that  their  sum  is  their 
strongest  common  consequence,  strong  and  weak  being  used 
in  a  sense  that  every  proposition  which  implies  another  is 
stronger  than  the  latter  and  the  latter  is  weaker  than  the 
one  which  implies  it.  Thus  it  is  easily  seen  that  the  product 
of  two  propositions  consists  in  their  simultaneous  affirmation: 
"a  and  b  are  true",  or  simply  "a  and  V ;  and  that  their 
sum  consists  in  their  alternative  affirmation,  "either  a  or  b 
is  true",  or  simply  "<z  or  b". 

Remark. — ^Logical  addition  thus  defined  is  not  disjunctive;^ 
that  is  to  say,  it  does  not  presuppose  that  the  two  summands 
have  no  element  in  common. 

8.  Principles  of  Simplification  and  Composition. — 
The  two  preceding  definitions,  or  rather  the  postulates  which 
precede  and  justify  them,  yield  directly  the  following  formulas : 

(i)  ab  <ia,  ab<^  b, 

(2)  {x<a){x<b)<{x<ab), 

(3)  a^a-^b,        b<^a-Vh, 

(4)  {a<x){b<x)<{a-^b<x). 

Formulas  (i)  and  (3)  bear  the  name  of  the  principle  of 
simplification  because  by  means  of  them  the  premises  of  an 

I  [Boole,  closely  following  analogy  with  ordinary  mathematics,  premised, 
as  a  necessary  condition  to  the  definition  of  "x  -\-  y",  that  x  and>'  were 
TOUtually  exclusive,  Jevons,  and  practically  all  mathematical  logicians  after 
him,  advocated,  on  various  grounds,  the  definition  of  "logical  addition" 
in  a  form  which  does  not  necessitate  matual  exclusiveness.] 


12  THE   LAWS    OF   TAUTOLOGY    AND    OF    ABSORPTION. 

argument  may  be  simplified  by  deducing  therefrom  weaker 
propositions,  either  by  deducing  one  of  the  factors  from  a 
product,  or  by  deducing  from  a  proposition  a  sum  (alter- 
native) of  which  it  is  a  summand. 

Formulas  (2)  and  (4)  are  called  the  principle  of  composition, 
because  by  means  of  them  two  inclusions  of  the  same  ante- 
cedent or  the  same  consequent  may  be  combined  {composed). 
In  the  first  case  we  have  the  product  of  the  consequents, 
in  the  second,  the  sum  of  the  antecedents. 

The  formulas  of  the  principle  of  composition  can  be  trans- 
formed into  equalities  by  means  of  the  principles  of  the 
syllogism  and  of  simplification.     Thus  we  have 

1  (Syll.)  {x  <  ab)  {ab  <  a)<  (^  <  a), 
(Syll.)  {X  <  ab)  {ab  <b)<(x<  b). 

Therefore 

(Comp.)  (x<ab)<(x<a)  (x<b). 

2  (Syll.)  (a<a  +  b)  (a  +  b<xX(a<x), 
(Syll.)  {b<a  +  b)  {a  +  b<x)<{b<x). 

Therefore 

(Comp.)  (a  +  b<xX(a<:x)  (b<.x). 

If  we    compare    the    new    formulas   with  those   preceding, 
which  are  their  converse  propositions,  we  may  write 
(x<ab)==(x<a)  {x<b), 
{a  +  b <^x)  =  {a  <:^x)  (b  <C  x). 

Thus,  to  say  that  x  is  contained  in  ab  is  equivalent  to 
saying  that  it  is  contained  at  the  same  time  in  both  a  and  b; 
and  to  say  that  x  contains  a  +  b  is  equivalent  to  saying 
that  it  contains  at  the  same  time  both  a  and  b, 

9.  The  Laws  of  Tautology  and  of  Absorption. — 
Since  the  definitions  of  the  logical  sum  and  product  do  not 
imply  any  order  among  the  terms  added  or  multiplied, 
logical  addition  and  multiplication  evidently  possess  commu- 
tative and  associative  properties  which  may  be  expressed  in 
the  formulas 

ab  =  ba,  I  a  -\-  b  =  b  +  a, 

(ab)  c  =  a  {be),      \  {a  + b)  +  c  ==  a  +  {b -^  c). 


THE   LAWS    OF    TAUTOLOGY    AND    OF    ABSORPTION.  1 3 

Moreover  they  possess  a  special  property  which  is  expressed 
in  the  law  of  tautology: 

a  =  aa^       \       a  ==  a  •\-  a. 

Demonstration  : 
I   (Simpl.)  aa  <C.  a, 

(Comp.)        (a  <^a)  (a<Ca)  =  {a<i  aa) 
whence,  by  the  definition  of  equality, 

(aa  <^a)  (a  <i  aa)  =  (a  =  aa). 
In  the  same  way: 
2   (Simpl.)  a<^a  +  a, 

(Comp.)      (a<ia)  (a  ■<  ^)  =  («  +  a  <C  «), 
whence 

(a-<Ca  +  a)  (a  +  a  <^  a)  =  (a  -^  a  +  a). 

From  this  law  it  follows  that  the  sum  or  product  of  any 
number  whatever  of  equal  (identical)  terms  is  equal  to  one 
single  term.  Therefore  in  the  algebra  of  logic  there  are 
neither  multiples  nor  powers,  in  which  respect  it  is  very 
much  simpler  than  numerical  algebra. 

Finally,  logical  addition  and  multiplication  possess  a 
remarkable  property  which  also  serves  greatly  to  simplify 
calculations,  and  which  is  expressed  by  the  law  of  absorption: 
a  +  ab  =  a,       |       a  {a  ■\-  b)  =  a. 

Demonstration : 

1  (Comp.)      {a  <C  a)  {ab  <^  a)  <^  {a  ■\-  ab  <i  a), 
(Simpl.)  a<^a  -\-  ab, 

whence,  by  the  definition  of  equality, 

(a  +  ab  <^  a)  (a<Ca  +  ab)  =  (a  +  ab  =  a). 
In  the  same  way: 

2  (Comp.)     (a<Ca)  (a  <C  a  +  b)  <C[a  <l  a  (a  +  b)], 
(Simpl.)  a  (a  +  b)  <^  a, 

whence 

[a  <  «  (a  +  b)]  [a  (a  +  bX.a]  =  [a  (a  +  b)  =  a]. 

Thus  a  term  (a)  absorbs  a  summand  (ab)  of  which  it  is  a 
factor,  or  a  factor  (a  +  b)  of  which  it  is  a  summand. 


14  THEOREMS    ON    MULTIPLICATION    AND    ADDITION. 

lo.  Theorems    on   Multiplication    and  Addition. — We 

can  now  establish  two  theorems  with  regard  to  the  com- 
bination of  inclusions  and  equalities  by  addition  and  multi- 
plication: 

(Th.  I)      {a<b)<{ac<bc),     \     {a<bX{a^-c<b  +  c). 
Demonstration  : 

1  (Simpl.)  ac  <i  c, 

(Syll.)  {ac <a)  {a<b)<  {ac <  b), 

(Comp.)  {ac  <^  b)  {ac  <C  ^)  <C  (^'^  <C  ^^)' 

2  (Simpl.)  c  <Cb  -\-  c, 
(Syll.)              {a<b)  {b<b-Vc)<{a<b-^c), 
(Comp.)  (a  <  3  +  <r)  {c  <Cb  -\-  c)<.{a  -\-  c  <^  b  -\-  c). 

This    theorem   may    be    easily    extended    to    the    case    of 
equalities: 

(a  =  ^)<  {ac  =  bc),     I     («  =  <J)<  («  +  <:  =  ^  +  c). 
(Th.  n)  {a<b)  {c<d)<.  {ac  <  bd), 

{a<:b)  {c<d)<{a  +  c<b  +  d). 
Demonstration : 

1  (Syll.)  {ac  <a)  {a<bX  {ac  <  b), 
(Syll.)  {ac<c)  {c<d)<{ac<d), 
(Comp.)                  {ac <^b)  {ac<.d)<Z  {ac <  bd). 

2  (Syll.)  {a<b)  {b<b  +  d)<{a<b  +  d), 
(Syll.)                {c<d)  {d<b  +  d)<{c<b  +  d)> 
(Comp.)     {a<^b  +  d)   (c  <  b  +  d)<.  {a  +  c  <:  b  +d). 

This  theorem  may  easily  be  extended  to  the  case  in  which 
one  of  the  two  inclusions  is  replaced  by  an  equality: 
{a  ==3)  (^  <  ^)<  {ac  <  bd), 
{a^b)  {c<:dX{a  +  c<b  +  d). 
When   both    are   replaced   by    equalities    the   result  is   an 
equality: 

{a  =  b)  {c=  d)<C  {ac  =  bd), 
{a  =  b)  {c^  dX.(a  +  c  =  b  +  cT}- 
To  sum  up,    two   or  more  inclusions  or  equalities   can  be 
added  or  multiplied  together  member  by  member;  the  result 
will  not  be  an  equality  unless  all  the  propositions  combined 
are  equalities. 


TRANSFORMING   INCLUSIONS.  1$ 

II.  The  First  Formula  for  Transforming  Inclusions 
into  Equalities. — We  can  now  demonstrate  an  important 
formula  by  which  an  inclusion  may  be  transformed  into  an 
equality,  or  vice  versa: 

(a<:i>)  =  (a  =  ad)       \       {a<^b)  ^  {a  +  b  =  b) 

Demonstration  : 

1.  (a  <  <J)<  (a  =  ab),  {a<bX{a-\-b  =  b). 
For 

(Comp.)  (a<a)  (a<^)<(a<^^), 

{a<b)  {b<b)<{a-Vb<b). 
On  the  other  hand,  we  have 
(SimpL)  ab<Ca,  b<ia-\-  b, 

(Def.  =)  {a  <  ab)  {ab<^a)^{a  =  ab\ 

{a^-b<b)  (b<a  +  b)  =  (a  +  b  =  b); 

2.  {a  =  ab)<(a<:b),  (a  +  b  ^  b)<(a<b). 
For 

(a^ab)  (ab<b)<(a<b), 
(a<a  +  b)  {a  +  b  =  b)<.{2L<b). 
Remark. — If  we  take  the  relation  of  equality  as  a  primitive 
idea  (one  not  defined)  we  shall  be  able  to  define  the  relation 
of  inclusion  by  means  of  one  of  the  two  preceding  formulas.* 
We  shall  then  be  able  to  demonstrate  the  principle  of  the 
syllogism.* 

From  the  preceding  formulas  may  be  derived  an  inter- 
esting result: 

{a  =  b)  =  {ab  =^  a-\-  b). 
For 
I.  {a=^b)=^{a<b)  {b<a), 

(a<ib)  =  (a^=  ab),     (b<.a)  =  (a  +  b  =  a), 
(Syll.)  (a  =  ab)  (a  +  b  ==  a)  <^(ab  ==  a  +  b). 

1  See  Huntington,  op.  cit.,  %  i. 

2  This  can  be  demonstrated  as  follows:  By  definition  we  have 
(a<C._b)  =  {a-=ab),  and  {J)<ic)^{p^=bc).  If  in  the  first  equality  we 
substitute  for  b  its  value  derived  from  the  second  equality,  then  a  =  abc. 
Substitute  for  a  its  equivalent  ab,  then  ab=^abc.  This  equality  is 
equivalent  to  the  inclusion,  ab  <^  c.  Conversely  substitute  a  for  ab; 
whence  we  have  a  <  <r.     q.  e.  d. 


l5  DISTRIBUTIVE   LAW. 

2.  {ab  =  a  +  b)<^{a  +  b<iab), 

(Comp.)         (a  +  ^  <  ar<5)  =  (a  <  ab)  {b  <  ab), 

{a  <  ab)  {ab  <^a)  ==  (a  ^  ab)  =^  {a  <  b), 
{b  <  ab)  {ab  <:b)  =  {b  =  ab)  =  {b  <  a). 

Hence 

{ab^a  +  b)<{a<  b)  {b^a)  =  {a  =  b). 

12.  The  Distributive  Law, — The  principles  previously 
stated  make  it  possible  to  demonstrate  the  converse  distributive 
law,  both  of  multiplication  with  respect  to  addition,  and  of 
addition  with  respect  to  multiplication, 

ac  +  be  <  (a  +  b)c,         ab^-c<.{a-]r  c)  {b -\-  c). 

Demonstration  : 

{a  ^a  ■{■  b)  <^[ac  <.{a  +  b)c], 
{b<a-^b)<:[bc<,{a-\-b)c]- 

whence,  by  composition, 

{ac<i{a  +  b)c]  [be  <  (a  +  b)e]  <  [ac  +  ^^  <  (a  +  b)e\ 

2.  {ab  <i  a)  <^  {ab  +  c  <i  a  +  c), 

{ab<bX{ab  +  c<b  +  e), 
whence,  by  composition, 
{ab+e<Za  +  e)  {ab  +  e <^  b  +  e)<i[ab  +  c <^  {a  +  e)  {b  +  e)]. 

But  these  principles  are  not  sufficient  to  demonstrate  the 
direct  distributive  law 

{a-\-b)c<^ae\bc,         {a -^  c)  {b  ^  e)<^ab -\-  c, 

and  we  are   obliged  to   postulate   one   of  these   formulas  or 
some   simpler   one    from    which   they    can   be  derived.     For 
greater  convenience  we  shall  postulate  the  formula 
(Ax.  V).  {a  -{■  b)c<^ac-\-  be. 

This,  combined  with  the  converse  formula,  produces  the 
equality 

{a  +  b)  e  =  ac  ■{■  be, 
which  we  shall  call  briefly  the  distributive  laiv. 
From  this  may  be  directly  deduced  the  formula 
{a\b)  {e-\-d)  =  ae-V  be-\-ad-\r  bd, 


DEFINITION    OF    O    AND    I.  1/ 

and  consequently  the  second  formula  of  the  distributive  law, 

(a  +  c)  (d  +  c)  =  ab-\-  c. 
For 

{a  ■\-  c)  {b  ■\-  c)  =  ab  ^r  ac  -\r  be  ^  c, 
and,  by  the  law  of  absorption, 

ac  ■\-  be  +  c  =  c. 

This  second  formula  implies  the  inclusion  cited  above, 

{a  +  c)  {b-\-c)<^ab  +  e, 

which  thus  is  shown  to  be  proved. 

Corollary. — We  have  the  equaHty 

ab  ■\-  ae  -^^  be  ^=  {a  +  b)  {a  -\-  e)  {b  -\-  e), 
for 

{a  +  ^)  (a  +  e)  {b  ■\-  e)  -=  {a  ■\- b e)  {b  ■{■  e)  =-  a b  ■\-  a c -\-  be. 

It  will  be  noted  that  the  two  members  of  this  equality 
differ  only  in  having  the  signs  of  multiplication  and  addition 
transposed  (compare  %  14). 

13.  Definition  of  o  and  i. — We  shall  now  define  and 
introduce  into  the  logical  calculus  two  special  terms  which 
we  shall  designate  by  o  and  by  i,  because  of  some  formal 
analogies  that  they  present  with  the  zero  and  unity  of  arith- 
metic. These  two  terms  are  formally  defined  by  the  two 
following  principles  which  affirm  or  postulate  their  existence. 

(Ax.  VI).  There  is  a  term  o  such  that  whatever  value 
may  be  given  to  the  term  x,  we  have 

(Ax.  VII).  There  is  a  term  i  such  that  whatever  value 
may  be  given  to  the  term  x,  we  have 

X  <Ct^. 

It  may  be  shown  that  each  of  the  terms  thus  defined  is 
unique;  that  is  to  say,  if  a  second  term  possesses  the  same 
property  it  is  equal  to  (identical  with)  the  first. 


l8  DEFINITION    OF    O    AND    I. 

The  two  interpretations  of  these  terms  give  rise  to  para- 
doxes which  we  shall  not  stop  to  elucidate  here,  but  which 
will  be  justified  by  the  conclusions  of  the  theory.' 

C.  I.:  o  denotes  the  class  contained  in  every  class;  hence 
it  is  the  "null"  or  "void"  class  which  contains  no  element 
(Nothing  or  Naught),  i  denotes  the  class  which  contains  all 
classes;  hence  it  is  the  totality  of  the  elements  which  are 
contained  within  it.  It  is  called,  after  Boole,  the  "universe 
of  discourse"  or  simply  the  "whole". 

P.  I.:  o  denotes  the  proposition  which  implies  every  prop- 
osition; it  is  the  "false"  or  the  "absurd",  for  it  implies 
notably  all  pairs  of  contradictory  propositions,  i  denotes 
the  proposition  which  is  implied  in  every  proposition;  it  is 
the  "true",  for  the  false  may  imply  the  true  whereas  the  true 
can  imply  only  the  true. 

By  definition  we  have  the  following  inclusions 

o<o,        o<i,        I  <  I, 

the   first    and  last  of  which,   moreover,  result  from  the  prin- 
ciple of  identity.     It  is  important  to  bear  the  second  in  mind. 

C.  I.:  The  null  class  is  contained  in  the  wholes 

P.  I.:  The  false  implies  the  true. 

By  the  definitions  of  o  and   i  we  have  the  equivalences 

(«  <  o)  =  (a  =  o),       (i  <;  a)  =  (a  =  i), 

since  we  have 

o  <C[  a,       a<^t 

whatever  the  value  of  a. 

Consequently  the  principle  of  composition  gives  rise  to 
the  two  following  corollaries: 

{a  ==  6)  {b  =  o)  =  {a  -V  b  =  o), 
{a  =  i)  (/5  =  i)  =  {ab  =  i). 

Thus  we  can  combine  two  equalities  having  o  for  a  second 

«  Compare  the  author's  Manuel  de  Logistique,  Chap.  I.,  %  8,  Paris, 
1905  [This  work,  however,  did  not  appear]. 

2  The  rendering  "Nothing  is  everything"  must  be  avoided. 


"y- 


DEFINITION    OF    O    AND    I.  1 9 

member  by  adding  their  first  members,  and  two  equalities 
having  i  for  a  second  member  by  multiplying  their  first 
members. 

Conversely,  to  say  that  a  sum  is  "null"  [zero]  is  to  say  that 
each  of  the  summands  is  null;  to  say  that  a  product  is  equal 
to  I  is  to  say  that  each  of  its  factors  is  equal  to  i. 
Thus  we  have 

(«  +  ^  =  o)<  (^  =  o), 
{ab  =  i)<^(a  =^  i), 
and  more  generally  (by  the  principle  of  the  syllogism) 
(a<d)  (3=o)<(a  =  o), 
(a <b)  {a=i)<{b=  i).  rc/v-w-*>J 

_^ „-         -    _    ,   -         -.     — Y'^^t 

ceptual  interpretation  the  first  equality  denotes  that  the  part  u^^^^")    ">r 
common    to    the    classes    a    and    b  is   null;    it  by  no  means  .  tr 
follows   that   either   one  or  the  other  of  these  classes  is  null.^ 
The    second    denotes  that  these  two   classes  combined  form 
the   whole;   it  by  no   means  follows  that  either  one   or  the 
other  is  equal  to  the  whole. 

The   following    formulas    comprising  the   rules  for  the  cal- 
culus of  o  and  i,  can  be  demonstrated: 

<a:Xo  =  o,       dr  +  I  =  I, 
a  +  o  =  a,       ay<i  =  a. 

For 

{o  <^  a)  =  {o  =  o  X  a)  ^  {a  -{-  o  ==  a), 

{a  <C  \)  =  {a  =  ax  i)  =  {a  -^  1  =  i). 

Accordingly  it  does  not  change  a  term  to  add  o  to  it  or 
to    multiply    it   by    i.     We   express  this  fact  by  saying  that 

0  is  the  modulus  of  addition  and  i  the  modulus  of  multi- 
plication. On  the  other  hand,  the  product  of  any  term 
whatever  by  o  is  o  and  the  sum  of  any  term  whatever  with 

1  is  I. 
These  formulas   justify    the    following   interpretation  of  the 

two  terms: 

2* 


20  DUALITY. 

C.  I.:  The  part  common  to  any  class  whatever  and  to  the 
null  class  is  the  null  class;  the  sum  of  any  class  whatever 
and  of  the  whole  is  the  whole.  The  sum  of  the  null  class  and 
of  any  class  whatever  is  equal  to  the  latter;  the  part  common 
to  the  whole  and  any  class  whatever  is   equal  to  the  latter. 

P.  I.:  The  simultaneous  affirmation  of  any  proposition 
whatever  and  of  a  false  proposition  is  equivalent  to  the  latter 
(i.  e.,  it  is  false);  while  their  alternative  affirmation  is  equal 
to  the  former.  The  simultaneous  affirmation  of  any  prop- 
osition whatever  and  of  a  true  proposition  is  equivalent  to 
the  former;  while  their  alternative  affirmation  is  equivalent  to 
the  latter  (i.  e.,  it  is  true). 

Remark. — If  we  accept  the  four  preceding  formulas  as 
axioms,  because  of  the  proof  afforded  by  the  double  inter- 
pretation, we  may  deduce  from  them  the  paradoxical  formulas 

o  <^x^    and    x  <^\^ 

by  means  of  the  equivalences  established  above, 

{a  =-  ab)  ^  {a  <ib)  =  {a  ■\-  b  =  b). 

14.  The  Law  of  Duality. — We  have  proved  that  a  perfect 
symmetry  exists  between  the  formulas  relating  to  multiplication 
and  those  relating  to  addition.  We  can  pass  from  one  class 
to  the  other  by  interchanging  the  signs  of  addition  and 
multiplication,  on  condition  that  we  also  interchange  the 
terms  o  and  i  and  reverse  the  meaning  of  the  sign  <^  (or 
transpose  the  two  members  of  an  inclusion).  This  symmetry,  or 
duality  as  it  is  called,  which  exists  in  principles  and  definitions, 
must  also  exist  in  all  the  formulas  deduced  from  them  as 
long  as  no  principle  or  definition  is  introduced  which  would 
overthrow  them.  Hence  a  true  formula  may  be  deduced 
from  another  true  formula  by  transforming  it  by  the  principle 
of  duality;  that  is,  by  following  the  rule  given  above.  In  its 
application  the  law  of  duality  makes  it  possible  to  replace 
two  demonstrations  by  one.  It  is  well  to  note  that  this  law 
is  derived  from  the  definitions  of  addition  and  multipli- 
cation  (the    formulas    for    which    are    reciprocal   by    duality) 


DEFINITION    OF    NEGATION.  2.1 

and  not,  as  is  often  thought^,  from  the  laws  of  negation 
which  have  not  yet  been  stated.  We  shall  see  that  these 
laws  possess  the  same  property  and  consequently  preserve 
the  duality,  but  they  do  not  originate  it;  and  duality  would 
exist  even  if  the  idea  of  negation  were  not  introduced.  For 
instance,  the  equality  (§12) 

ab-\-  ac^-bc={a-V  b)  {a  +  c)  (b  +  c) 
is    its    own   reciprocal    by  duality,    for    its  two   members  are 
transformed  into  each  other  by  duality. 

It  is  worth  remarking  that  the  law  of  duality  is  only 
applicable  to  primary  propositions.  We  call  [after  Boole] 
those  propositions  primary  which  contain  but  one  copula 
(<C  or  ==).  We  call  those  propositions  secondary  of  which 
both  members  (connected  by  the  copula  <![  or  =)  are  primary 
propositions,  and  so  on.  For  instance,  the  principle  of 
identity  and  the  principle  of  simplification  are  primary  pro- 
positions, while  the  principle  of  the  syllogism  and  the  principle 
of  composition  are  secondary  propositions. 

15.  Definition  of  Negation.— The  introduction  of  the  terms 
o  and  I  makes  it  possible  for  us  to  define  negation.  This 
is  a  "uni-naiy"  operation  which  transforms  a  single  term  into 
another  term  called  its  negative.^  The  negative  of  a  is  called 
not-a  and  is  written  <z'.j  Its  formal  definition  implies  the 
following  postulate  of  existence  "♦: 

I  [Boole  thus  derives  it  {Laws  of  Thought,  London  1854,  Chap.  Ill, 
Prop.  IV).] 

»  [In  French]  the  same  word  negation  denotes  both  the  operation 
and  its  result,  which  becomes  equivocal.  The  result  ought  to  be  denoted 
by  another  word,  like  [the  English]  "negative".  Some  authors  say,  "supple- 
mentary" or  "supplement",  [e.  g.  Boole  and  Huntington].  Classical 
logic  makes  use  of  the  term  "contradictory"  especially  for  propositions. 

3  We  adopt  here  the  notation  of  MacColl;  SchrOder  indicates 
not-a  by  a  I  which  prevents  the  use  of  indices  and  obliges  us  to  express 
them  as  exponents.  The  notation  a'  has  the  advantage  of  excluding 
neither  indices  nor  exponents.  The  notation  a  employed  by  many 
authors  is  inconvenient  for  typographical  reasons.  "When  the  negative 
afTects  a  proposition  written  in  an  explicit  form  (with  a  copula)  it  is 
applied  to  the  copula  «  or  ==)  by  a  vertical  bar  {<^  or  =}=)•  The 
accent  can  be  considered  as  the  indication  of  a  vertical  bar  applied  to  letters. 

4  [Boole   follows  Aristotle  in  usually  calling  the   law   of  duality  the 


22  DEFINITION    OF   NEGATION. 

(Ax.  Vin.)  Whatever  the  term  a  may  be,  there  is  also  a 
term  a    such  that  we  have  at  the  same  time 

a  a  =o,     a-\-a'  =  i. 

It  can  be  proved  by  means  of  the  following  lemma  that  if 
a  term  so  defined  exists  it  is  unique: 
If  at  the  same  time 

ac  =  be,     a  -{-  c  =  b  ■\-  c, 
then 

a  ==  b. 

Demonstration. — Multiplying  both  members  of  the  second 
premise  by  a,  we  have 

a  -^^  ac  ^=  ab  -\-  ac. 
Multiplying  both  members  by  b, 

ab  -\-  be  =  b  Ar  be. 

By  the  first  premise, 

ab  -Y  ae  ==  ab  -\-  be. 
Hence 

a-\-  ac  =  b  ■\-  be, 

which  by  the  law  of  absorption  may  be  reduced  to 

a  =  b. 

Remark. — This  demonstration  rests  upon  the  direct  dis- 
tributive law.  This  law  cannot,  then,  be  demonstrated  by  means 
of  negation,  at  least  in  the  system  of  principles  which  we  are 
adopting,  without  reasoning  in  a  circle. 

This  lemma  being  established,  let  us  suppose  that  the  same 
term  a  has  two  negatives;  in  other  words,  let  «'i  and  ci  ^  be 
two  terms   each  of  which  by  itself  satisfies  the  conditions  of 


principle  of  contradiction  "which  affirms  that  it  is  impossible  for  any 
being  to  possess  a  quality  and  at  the  same  time  not  to  possess  it".  He 
writes  it  in  the  form  of  an  equation  of  the  second  degree,  x  —  x'^  =  o, 
OT  X  (I  — jr)  =  o  in  which  I  — x  expresses  the  universe  less  x,  or  not 
•  X.  Thus  he  regards  the  law  of  duality  as  derived  from  negation  as 
stated  in  note  i  above.] 


CONTRADICTION  AND  EXCLUDED    MIDDLE.  23 

the  definition.  We  will  prove  that  they  are  equal.  Since, 
by  hypothesis, 

ad  2  =  0,     a  +  a'2  =  I , 
we  have 

aa^  =  aa^,     a -V  o,  x=^  (i-\- a^'t 

whence  we  conclude,  by  the  preceding  lemma,  that 

a\  =  a  2 . 

We  can  now  speak  of  //5<?  negative  of  a  term  as  of  a  unique 
and  well-defined  term. 

The  uniformity  of  the  operation  of  negation  may  be  ex- 
pressed in  the  following  manner: 

If  a  =  b,  then  also  a  ^=  b' .  By  this  proposition,  both 
members  of  an  equality  in  the  logical  calculus  may  be 
"denied", 

16.  The  Principles  of  Contradiction  and  of  Excluded 
Middle. — By  definition,  a  term  and  its  negative  verify  the 
two  formulas 

ad  ^  o,     a  +  a'  =  I , 

which  represent  respectively  the  principle  of  contradiction  and 
the  principle  of  excluded  middle.^ 

C.  I;  I.  The  classes  a  and  d  have  nothing  in  common; 
in  other  words,  no  element  can  be  at  the  same  time  both  a 
and  not-a. 

2,  The  classes  a  and  d  combined  form  the  whole;  in 
other  words,  every  element  is  either  a  or  not-«. 


I  As  Mrs.  Ladd-Franklin  has  truly  remarked  (Baldwin,  Dictionary 
of  Philosophy  and  Psychology,  article  "Laws  of  Thought"),  the  principle  of 
contradiction  is  not  sufficient  to  define  contradictories ;  the  principle  of 
excluded  middle  must  be  added  which  equally  deserves  the  name  of 
principle  of  contradiction.  This  is  why  Mrs.  Ladd-Franklin  proposes 
to  call  them  respectively  the  principle  of  exclusion  and  the  principle  of 
exhaustion,  inasmuch  as,  according  to  the  first,  two  contradictory  terms 
are  exclusive  (the  one  of  the  other);  and,  according  to  the  second,  they 
are  exhaustive  (of  the  universe  of  discourse). 


24  DOUBLE   NEGATION. 

P.  I.:  I.  The  simultaneous  affirmation  of  the  propositions 
a  and  not-a  is  false;  in  other  words,  these  two  propositions 
cannot  both  be  true  at  the  same  time. 

2.  The  alternative  affirmation  of  the  propositions  a  and 
not-a  is  true;  in  other  words,  one  of  these  two  propositions 
must  be  true. 

Two  propositions  are  said  to  be  contradictory  when  one  is 
the  negative  of  the  other;  they  cannot  both  be  true  or  false 
at  the  same  time.  If  one  is  true  the  other  is  false;  if  one 
is  false  the  other  is  true. 

This  is  in  agreement  with  the  fact  that  the  terms  o  and  i 
are  the  negatives  of  each  other;  thus  we  have 

oxi  =  o,     o+i  =  i. 

Generally  speaking,  we  say  that  two  terms  are  contradictory 
when  one  is  the  negative  of  the  other. 

17.  Lav7  of  Double  Negation. — Moreover  this  reciprocity 
is  general:  if  a  term  b  is  the  negative  of  the  term  a,  then  the 
term  a  is  the  negative  of  the  term  b.  These  two  statements 
are  expressed  by  the  same  formulas 

ab  =  o,     a  +  ^  ==  I, 

and,  while  they  unequivocally  determine  b  in  terms  of  a,  they 
likewise  determine  a  in  terms  of  b.  This  is  due  to  the 
symmetry  of  these  relations,  that  is  to  say,  to  the  commu- 
tativity  of  multiplication  and  addition.  This  reciprocity  is 
expressed  by  the  law  of  double  negation 

{a')'  =  a, 

which  may  be  formally  proved  as  follows:  ci  being  by  hy- 
pothesis the  negative  of  a,  we  have 

aa  =  o,     a  +  a'  =  1. 
On  the  other  hand,  let  a"  be  the  negative  of  a';  we  have, 
in  the  same  way, 

a'a"  =  o,     a' +  a"  =  I. 
But,  by  the  preceding  lemma,  these  four  equalities  involve 
the  equality 

a  =  a".  Q.  E.  D. 


TRANSFORMING   INCLUSIONS.  25 

This  law  may  be  expressed  in  the  following  manner: 

If  ^  =  a',  we  have  a  =  b\  and  conversely,  by  symmetry. 

This  proposition  makes  it  possible,  in  calculations,  to 
transpose  the  negative  from  one  member  of  an  equality  to 
the  other. 

The  law  of  double  negation  makes  it  possible  to  conclude 
the  equality  of  two  terms  from  the  equality  of  their  negatives 
(if  a  =  b'  then  a  =  ^),  and  therefore  to  cancel  the  negation 
of  both  members  of  an  equality. 

From  the  characteristic  formulas  of  negation  together  with 
the  fundamental  properties  of  o  and  i,  it  results  that  every 
product  which  contains  two  contradictory  factors  is  null,  and 
that  every  sum  which  contains  two  contradictory  summands 
is  equal  to   i. 

In  particular,  we  have  the  following  formulas: 

a  =  ab  -^r  ab' ,     a  =  (a  i-  b)  (a  +  b'), 

which    may   be    demonstrated    as    follows   by   means    of  the 
distributive  law: 

a  =  axi  =  a(b  +  b')  =  ab  +  ab', 

a  =  a  +  o-=a  +  bb'    =(a  +  b)  (a  +  b') . 

These  formulas  indicate  the  principle  of  the  method  of 
development  which  we  shall  explain  in  detail  later  (§§  2 1  sqq.) 

18.  Second  Formula  for  Transforming  Inclusions 
into  Equalities: — We  can  now  establish  two  very  important 
equivalences  between  inclusions  and  equalities: 

(a<^)  =  (dr^'=  o),     (a<^)  =  (a'  +  ^=  i). 

Demonstration. — i.  If  we  multiply  the  two  members  of  the 
inclusion  a<Cb  by  b'  we  have 

{ab'  <  bb')  =  {ab'<^  o)  =  {ab'  =  o). 

2.  Again^  we  know  that 

a  =  ab  +  ab'. 
Now  if  ab'  =  o, 

a  =  ab  +  o  '=  ab. 


26  LAW    OF    CONTRAPOSITION, 

On  the  other  hand:  i.  Add  a  to  each  of  the  two  members 
of  the  inclusion  a<Cb\  we  have 

(a'  +  a  <  <z'  +  ^)  =  (i  <  a'  +  <5)  =  (a'  +  /^  =  i). 

2.  We  know  that 

b  =  {a-\-b)  {a  +d). 

Now,  if  a'  +  /5  ==  I , 

b=={a  +  b)xi=a  +  b. 

By  the  preceding  formulas,  an  inclusion  can  be  transformed 
at  will  into  an  equality  whose  second  member  is  either  o  or  i. 
Any  equality  may  also  be  transformed  into  an  equality  of 
this  form  by  means  of  the  following  formulas: 

(a  =  l>)  =  (ad'  +  a[d  =  o),    (a  =  b)=[(a  +  d')  {a'  +  b)=-i]. 

Demonstration  : 
{a  =  b)  =  {a<^b)  {b<^d)=-{ab' =6)  {a'b  =  o)  =  {ab' ■\-a'b=o), 

^a  =  b)  =  {a<b)  {b<:a)  =  (a  +b=^  i)  (a  +  b' =  i)  = 
[(^b')ia+b)==i]. 

Again,  we  have  the  two  formulas 

(a=b)  =  [(a  +  b)  {a  +  b')  =  o],     (a  =  3)  =  {ab  +  a  b'  =  i), 

which  can  be  deduced  from  the  preceding  formulas  by  per- 
forming the  indicated  multiplications  (or  the  indicated  additions) 
by  means  of  the  distributive  law, 

ig.  Law  of  Contraposition. — We  are  now  able  to  demon- 
strate the  law  of  contraposition^ 

{a<b)=^{b'<a'). 
Demonstration. — By  the  preceding  formulas,  we  have 

(a  < I,)  =  {ab'  =  o)  =  {b'  < a) , 
Again,  the  law  of  contraposition  may  take  the  form 

{a<:b')=={b<d), 

which  presupposes  the  law  of  double  negation.  It  may  be 
expressed  verbally  as  follows:  "Two  members  of  an  inclusion 
may  be  interchanged  on  condition  that  both  are  denied". 


POSTULATE    OF    EXISTENCE.  2"] 

C.  I.:  "If  all  a  is  b,  then  all  not-/J  is  not-a,  artd  conversely". 

P.  L:  "If  a  implies  b,  xio\.-b  implies  not-a  and  conversely"; 
in  other  words,  "If  a  is  true  b  is  true",  is  equivalent  to 
saying,  "If  b  is  false,  a  is  false". 

This  equivalence  is  the  principle  of  the  reductio  ad  absurdum 
(see  hypothetical  arguments,  modus  tol/ens,  %  58). 

20.  Postulate  of  Existence. — One  final  axiom  may  be 
formulated  here,  which  we  will  call  the  postulate  of  existence. 

(Ax.  IX)  i<o, 

whence  may  be  also  deduced  i  =H  o  • 

In  the  conceptual  interpretation  (C.  I.)  this  axiom  means 
that  the  universe  of  discourse  is  not  null,  that  is  to  say,  that 
it  contains  some  elements,  at  least  one.  If  it  contains  but 
one,  there  are  only  two  classes  possible,  i  and  o.  But  even 
then  they  would  be  distinct,  and  the  preceding  axiom  would 
be  verified. 

In  the  prepositional  interpretation  (P.  I.)  this  axiom  signifies 
that  the  true  and  the  false  are  distinct ;  in  this  case,  it  bears 
the  mark  of  evidence  and  of  necessity.  The  contrary 
proposition,  1=0,  is,  consequently,  the  type  of  absurdity 
(of  the  formally  false  proposition)  while  the  propositions  0  =  0, 
and  I  ^  I  are  types  of  identity  (of  the  formally  true  pro- 
position).    Accordingly  we  put 

(i  =  o)  =  o,     (0  =  0)  =  (I  =  i)  =  1. 

'  More  generally,  every  equality  of  the  form 

X  ^=  X 
is   equivalent  to  one  of  the  identity- types;   for,  if  we  reduce 
this  equality  so  that  its  second  member  will  be  o  or  i,  we  find 

{XX  +  XX  =  o)  =  (o  =  o),     {xx  +  x  x  =  i)  =  (i  =  i). 
On  the  other  hand,  every  equality  of  the  form 

X  =  X 

is  equivalent  to  the  absurdity-type,   for  we  find  by  the  same 
process, 

{xx  +  x  X  =  o)  =  (i  =  o),     {xx  +  XX  ==  i)  =  (o  =  1). 


28  DEVELOPMENTS    OF    O    AND    OF    I. 

21.  The  Developments  of  o  and  of  i. — Hitherto  we 
have  met  only  such  formulas  as  directly  express  customary 
modes   of  reasoning  and   consequently  offer  direct  evidence. 

We  shall  now  expound  theories  and  methods  which  depart 
from  the  usual  modes  of  thought  and  which  constitute  more 
particularly  the  algebra  of  logic  in  so  far  as  it  is  a  formal 
and,  so  to  speak,  automatic  method  of  an  absolute  universality 
and  an  infallible  certainty,  replacing  reasoning  by  cal- 
culation. 

The  fundamental  process  of  this  method  is  developfnent. 
Given  the  terms  a,  b,  c  .  .  .  (to  any  finite  number),  we  can 
develop  o  or  i  with  respect  to  these  terms  (and  their  negatives) 
by  the  following  formulas  derived  from  the  distributive  law: 

o  =^  aa\ 

o  =  aa'-\-  bb'  =  {a  -\-b)  {a-^  b')  {a -\- b)  {a -\-  b'), 

0  =  ad -\-  bb' -\-cc'  =  {a  ^r  b-\-  c)  (a  +  b  +  c)  (a  +  b'  +  c) 

X{a^-  b'  ■\-  c)  (a'+  b  +  c) 

X  (a'  +  3  +  /)  {a  +  b'-\-  c)  {a  +3  +  /); 

1  =  a  +  a', 

I  =  (a  +  a)  {b  +  b')  =  ab  +  ab'  +  a  b  +  a' b' , 

I  =  (a  +  d)  {b  +  b')  {c  +  /)  =  abc  +  abc  +  ab' c  +  ab' c 

+  a  be  +  a'bc  -\-  a' b'c  +  a'b'c'\ 

and  so  on.     In  general,   for  any  number  n  of  simple  terms 

0  will  be  developed  in  a  product  containing  2"  factors,  and 

1  in  a  sum  containing  2"  summands.  The  factors  of  zero 
comprise  all  possible  additive  combinations,  and  the  summands 
of  I  all  possible  multiplicative  combinations  of  the  n  given 
terms  and  their  negatives,  each  combination  comprising  n 
different  terms  and  never  containing  a  term  and  its  negative 
at  the  same  time. 

The  summands  of  the  development  of  i  are  what  Boole 
called  the  constituents  (of  the  universe  of  discourse).  We  may 
equally  well  call  them,  with  Poretsky,*  the  minima  of  dis- 
course, because   they  are  the  smallest  classes  into  which  the 

I  See  the  Bibliography,  page  xiv. 


PROPERTIES    OF   THE    CONSTITUENTS.  29 

universe  of  discourse  is  divided  with  reference  to  the  n  given 
terras.  In  the  same  way  we  shall  call  the  factors  of  the 
development  of  o  the  maxima  of  discourse,  because  they 
are  the  largest  classes  that  can  be  determined  in  the  universe 
of  discourse  by  means  of  the  n  given  terms. 

22.  Properties  of  the  Constituents. — The  constituents 
or  minima  of  discourse  possess  two  properties  characteristic 
of  contradictory  terms  (of  which  they  are  a  generalization); 
they  are  mutually  exclusive^  i.  e.,  the  product  of  any  two  of 
them  is  o;  and  they  are  collectively  exhaustive,  i.  e.,  the  sum 
of  all  "exhausts"  the  universe  of  discourse.  The  latter  prop- 
erty is  evident  from  the  preceding  formulas.  The  other 
results  from  the  fact  that  any  two  constituents  diifer  at  least 
in  the  "sign"  of  one  of  the  terms  which  serve  as  factors,  i.  e., 
one  contains  this  term  as  a  factor  and  the  other  the  negative 
of  this  term.  This  is  enough,  as  we  know,  to  ensure  that 
their  product  be  null. 

The  maxima  of  discourse  possess  analogous  and  correlative 
properties;  their  combined  product  is  equal  to  o,  as  we  have 
seen;  and  the  sum  of  any  two  of  them  is  equal  to  i,  inasmuch 
as  they  differ  in  the  sign  of  at  least  one  of  the  terms  which 
enter  into  them  as  summands. 

For  the  sake  of  simplicity,  we  shall  confine  ourselves,  with 
Boole  and  Schroder,  to  the  study  of  the  constituents  or 
minima  of  discourse,  /.  e.,  the  developments  of  i.  We  shall 
leave  to  the  reader  the  task  of  finding  and  demonstrating 
the  corresponding  theorems  which  concern  the  maxima  of 
discourse  or  the  developments  of  o. 

23.  Logical  Functions. — We  shall  call  a  logical  function 
any  term  whose  expression  is  complex,  that  is,  formed  of 
letters  which  denote  simple  terms  together  with  the  signs  of 
the  three  logical  operations.^ 


I  In  this  algebra  the  logical  function  is  analogous  to  the  integral 
funclion  of  ordinary  algebra,  except  that  it  has  no  powers  beyond 
the  first. 


30  LOGICAL   FUNCTIONS    AND    THEIR    DEVELOPMENT. 

A  logical  function  may  be  considered  as  a  function  of  all 
the  terms  of  discourse,  or  only  of  some  of  them  which  may 
be  regarded  as  unknown  or  variable  and  which  in  this  case 
are  denoted  by  the  letters  x,  y,  z.  We  shall  represent  a 
function  of  the  variables  or  unknown  quantities,  x,  y,  z,  by 
the  symbol  /  (x,  y,  z)  or  by  other  analogous  symbols,  as  in 
ordinary  algebra.  Once  for  all,  a  logical  function  may  be 
considered  as  a  function  of  any  term  of  the  universe  of  dis- 
course, whether  or  not  the  term  appears  in  the  explicit  ex- 
pression of  the  function. 

24.  The  Law  of  Development. — This  being  established, 
we  shall  proceed  to  develop  a  function  /{x)  with  respect  to  x. 
Suppose  the  problem  solved,  and  let 

ax  +  bx 

be    the   development    sought.      By    hypothesis    we    have    the 
equality 

/{x)  ^  ax  -^  bx' 

for  all  possible  values  of  x.     Make  .^  =  i   and   consequently 
x  =  o.     We  have 

/(i)  =  a. 

Then  put  x  =  o  and  x  ^  i]  we  have 

/(o)  =  b. 

These  two  equalities  determine  the  coefficients  a  and  b  of 
the  development  which  may  then  be  written  as  follows: 

f{x)^f{i)x+/{o)x\ 

in  which /(i),  /(o)  represent  the  value  of  the  function /(.ar) 
when  we  let  x  =  1    and  x  =  o  respectively. 

Corollary. — Multiplying  both  members  of  the  preceding 
equalities  by  x  and  x  in  turn,  we  have  the  following  pairs 
of  equalities  (Mac Coll): 

^/{^)  •=  ^x  x'f{x)  =  bx 

xfix)  =  x/{i),     x'f{x)  =  x'/{o) . 
Now  let  a  function  of  two  (or  more)  variables  be  developed 


LAW    OF    DEVELOPMENT.  3 1 

with    respect    to    the    two    variables    x    and  y.     Developing 
/(x,  y)  first  with  respect  to  x,  we  find 

fix,  y)=Aj,  y)x-^f{o,  y)x. 

Then,   developing  the  second   member  with  respect  to  y, 
we  have 
/(x,  y)=/{'L,i)xy-\-/{i,  o)^/+/(o,  i)x'y  +/(o,  o)xy'. 

This  result  is  symmetrical  with  respect  to  the  two  variables, 
and  therefore  independent  of  the  order  in  which  the  develop- 
ments with  respect  to  each  of  them  are  performed. 

In  the  same  way  we  can  obtain  progressively  the  develop- 
ment of  a  function  of  3,  4, ,  variables. 

The  general  law  of  these  developments  is  as  follows: 

To  develop  a  function  with  respect  to  n  variables,  form  all 
the  constituents  of  these  n  variables  and  multiply  each  of 
them  by  the  value  assumed  by  the  function  when  each  of 
the  simple  factors  of  the  corresponding  constituent  is  equated 
to  I  (which  is  the  same  thing  as  equating  to  o  those  factors 
whose  negatives  appear  in  the  constituent). 

When  a  variable  with  respect  to  which  the  development  is 
made*,  y  for  instance,  does  not  appear  explicitly  in  the 
function  {/{x)  for  instance),  we  have,  according  to  the 
general  law, 

fix)  =Ax)y  ■\-f{.x)y. 

In  particular,   if  a  is  a  constant  term,  independent  of  the 
variables  with   respect  to  which   the    development    is    made, 
we  have  for  its  successive  developments, 
a==  ax  -\-  ax\ 

a  =  axy  -t-  axy  -V  ax y  -H  ax'y, 
a  =  axyz  -f  axyz  +  axy  z  +  axy  z  +  ax'yz  +  ax yz  -t-  ax  y  z 

-h  ax'y'z'^ 
and  so  on.    Moreover  these  formulas  may  be  directly  obtained 
by  multiplying  by  a  both  members  of  each  development  of  i. 

Cor.  I.     We  have  the  equivalence 

{a  +  x)  {b  +  x)  ==  ax  -{■  bx  ■{■  ab  =  ax  +  bx. 

I  These  formulas  express  the  method  of  classificatioii  by  dichotomy. 


32  FORMULAS    OF   DE   MORGAN. 

For,  if  we  develop  with  respect  to  x,  we  have 
ax  +  dx  +  abx  +  abx'  ==  {a-\r  ab)x  +  (<^  +  ab)x'  =  ax  ■{■  bx. 

Cor.  2.     We  have  the  equivalence 

ax  +  bx'  +  r  ==  («  +  c)x  +  (<^  +  c)x'. 
For  if  we   develop   the  term  c  with  respect  to  x,  we  find 
ax  +  bx' -\-  ex  +  ex  ==  («  +  e)x  +  (<5  +  e)x' . 

Thus,  when  a  function  contains  terms  (whose  sum  is 
represented  by  e)  independent  of  x,  we  can  always  reduce  it 
to  .the  developed  form  ax-{-  bx  by  adding  e  to  the  coefficients 
of  both  X  and  x.  Therefore  we  can  always  consider  a 
function  to  be  reduced  to  this  form. 

In  practice,  we  perform  the  development  by  multiplying 
each  term  which  does  not  contain  a  certain  letter  {x  for 
instance)  by  {x  +  x)  and  by  developing  the  product  according 
to  the  distributive  law.  Then,  when  desired,  like  terms  may 
be  reduced  to  a  single  term. 

25.  The  Formulas  of  De  Morgan. — In  any  development 
of  /,  the  sum  of  a  certain  number  of  constituents  is  the  negative 
of  the  sum  of  all  the  others. 

For,  by  hypothesis,  the  sum  of  these  two  sums  is  equal 
to  I,  and  their  product  is  equal  to  o,  since  the  product  of 
two  diflferent  constituents  is  zero. 

From  this  proposition  may  be  deduced  the  formulas  of 
De  Morgan: 

{a  +  bf  =  ci  b' ,     {ab)  =  a  +  b' . 

Demonstration. — Let  us  develop  the  sum  {a-^b): 
a  +  b  ^  ab  ■{•  ab'  +  ab  +  ab  ==  ab  +  ab'  +  a  b. 

Now  the  development  of  i  with  respect  to  a  and  b  contains 
the  tliree  terms  of  this  development  plus  a  fourth  term  a  b' , 
This  fourth  term,  therefore,  is  the  negative  of  the  sum  of  the 
other  three. 

We  can  demonstrate  the  second  formula  either  by  a  correl- 
ative argument  {i.  e.,  considering  the  development  of  o  by 
factors)  or  by  observing  that  the  development  of  {a'-\-b'). 


FORMULAS    OF   DE   MORGAN.  33 

a  b  +  ab'  -^r  a  b\ 
differs  from  the  development  of  i   only  by  the  summand  ab. 
How    De  Morgan's    formulas  may   be   generalized  is  now 
clear;  for  instance  we  have  for  a  sum  of  three  terms, 

a  ■\-  b  ■\  c  '^  abc  +  abc  +  ab' c  +  ab' c  -^  a  bc-\-  a' be  -V  a' b'c. 

This  development  differs  from  the  development  of  i  only 
by  the   term  abc.     Thus  we  can   demonstrate  the  formulas 

{a  -\r  b  -V  c)'  =  a  b' c,     (abc)'  =  a  -^  b'  -\-  c, 

which  are  generalizations  of  De  Morgan's  formulas. 

The  formulas  of  De  Morgan  are  in  very  frequent  use  in 
calculation,  for  they  make  it  possible  to  perform  the  negation 
of  a  sura  or  a  product  by  transferring  the  negation  to  the 
simple  terms:  the  negative  of  a  sum  is  the  product  of  the 
negatives  of  its  summands;  the  negative  of  a  product  is  the 
sum  of  the  negatives  of  its  factors. 

These  formulas,  again,  make  it  possible  to  pass  from  a 
primary  proposition  to  its  correlative  proposition  by  duality, 
and  to  demonstrate  their  equivalence.  For  this  purpose  it 
is  only  necessary  to  apply  the  law  of  contraposition  to  the 
given  proposition,  and  then  to  perform  the  negation  of  both 
members. 

£xample: 

ab  -\-  ac  -\-  be  =  {a  ■\-  b)  {a  -\-  c)  {b  +  c). 
Demonstration  : 

{ab^  ac\  be)'  =  \{a  +  ^)  (a  +  c)  {b  +  0] , 
{ab)'  {ae)'{be)'  =  (a  +  b)'-V  (a  +  e)'  +  (^  +  c)\ 
(a  +  b')  {a  +  c)  {b'  +  /)  =  a  b'  +  a  e  +  b'e. 

Since  the  simple  terms,  a,  b,  e,  may  be  any  terms,  we  may 
suppress  the  sign  of  negation  by  which  they  are  affected,  and 
obtain  the  given  formula. 

Thus  De  Morgan's  formulas  furnish  a  means  by  which  to 
find  or  to  demonstrate  the  formula  correlative  to  another; 
but,  as  we  have  said  above  (%  14),  they  are  not  the  basis  of 
this  correlation. 

3 


34  DISJUNCTIVE    SUMS. 

26.  Disjunctive  Sums. — By  means  of  development  we  can 
transform  any  sum  into  a  disjunctive  sum,  /.  e.,  one  in  which 
each  product  of  its  summands  taken  two  by  two  is  zero. 
For,  let  {a-\-  b  ^r  c)  be  a  sum  of  which  we  do  not  know 
whether  or  not  the  three  terms  are  disjunctive;  let  us  assume 
that  they  are  not.     Developing,  we  have: 

a-\-  b  ■\-  c  =  abc  +  abc  -\-ab'  c  +  ab'  c  Ar  a  be  +  a  be  +  a  b'  c. 

Now,  the  first  four  terms  of  this  development  constitute 
the  development  of  a  with  respect  to  b  and  c;  the  two 
following  are  the  development  of  a  b  with  respect  to  c.  The 
above  sum,  therefore,  reduces  to 

a  +  a  b  -\-  a  b'  c , 

and  the  terms  of  this  sum  are  disjunctive  like  those  of  the 
preceding,  as  may  be  verified.  This  process  is  general  and, 
moreover,  obvious.  To  enumerate  without  repetition  all  the 
a's,  all  the  ^'s,  and  all  the  ^'s,  etc.,  it  is  clearly  sufficient  to 
enumerate  all  the  a's,  then  all  the  ^'s  which  are  not  a's,  and 
then  all  the  <r's  which  are  neither  a's  nor  ^'s,  and  so  on. 

It  will  be  noted  that  the  expression  thus  obtained  is  not 
symmetrical,  since  it  depends  on  the  order  assigned  to  the 
original  summands.     Thus  the  same  sum  may  be  written: 

b  ■\- ab' ^r  a  b' c ,     c  ■\- ac -\- a  bc\  .  .  .  . 

Conversely,  in  order  to  simplify  the  expression  of  a  sum, 
we  may  suppress  as  factors  in  each  of  the  summands  (arranged 
in  any  suitable  order)  the  negatives  of  each  preceding  sum- 
mand.  Thus,  we  may  find  a  symmetrical  expression  for  a 
sum.     For  instance, 

a  +  a  b  =  b  -\-  ab'  =  a  -\-  b . 

27.  Properties  of  Developed  Functions. — The  practical 
utility  of  the  process  of  development  in  the  algebra  of  logic 
lies  in  the  fact  that  developed  functions  possess  the  following 
property: 

The  sum  or  the  product  of  two  functions  developed  with 
respect  to  the  same  letters  is  obtained  simply  by  finding  the 
sum  or  the  product  of  their  coefficients.     The  negative  of  a 


PROPERTIES  OF  DEVELOPED  FUNCTIONS.  35 

developed  function  is  obtained  simply  by  replacing  the 
coefficients  of  its  development  by  their  negatives. 

We  shall  now  demonstrate  these  propositions  in  the  case 
of  two  variables;  this  demonstration  will  of  course  be  of 
universal  application. 

Let  the  developed  functions  be 

a^xy  +  bt.xy'  +  c^x  y  +  d^x'y, 
a 2  xy  +  ^2  xy  +  C2  x'y  +  ^2  x'y  . 

1.  I  say  that  their  sum  is 

(«!  +  a2)xy  +  {bi  +  ^2)^/+  {^i  +  C2)xy  +  {d^  +  d^xy. 
This  result  is  derived  directly  from  the  distributive  law. 

2.  I  say  that  their  product  is 

a^a^xy  +  b^b^xy  ^r  c-^c^x y  +  d^^d^x y\ 

for  if  we  find  their  product  according  to  the  general  rule 
(by  applying  the  distributive  law),  the  products  of  two  terms 
of  difierent  constituents  will  be  zero;  therefore  there  will  remain 
only  the  products  of  the  terms  of  the  same  constituent,  and, 
as  (by  the  law  of  tautology)  the  product  of  this  constituent 
multiplied  by  itself  is  equal  to  itself,  it  is  only  necessary  to 
obtain  the  product  of  the  coefficients. 

3.  Finally,  I  say  that  the  negative  of 

axy  ■\-  bxy  +  ex  y  +  dx'y 
is 

d  xy  +  b'  xy  +  c  x'y  +  d'  x'y. 

In  order  to  verify  this  statement,  it  is  sufficient  to  prove 
that  the  product  of  these  two  functions  is  zero  and  that  their 
sum  is  equal  to   i.     Thus 

{axy  +  bxy  +  cx'y  -f  dx'y)  {a  xy  +  b' xy  +  c  x'y  +  d' xy) 
=  {ad  xy  4-  bb'xy  +  cc  x'y  +  dd'  x'y) 

=  {o'xy^-\-  O'xy  -\-  O'x'y  -{■  o-x'y)=o 

(axy  +  bxy  +  cx'y  +  dxy)  +  {dxy  +  V  xy  +  c'xy  +  d'  x'y) 
=  [(a  +  d) xy-\-{b-^ b') xy  +  (r + /) xy  ^{d-\-d) x'y] 

=  {i-xy+  I  -xy  +  I  -x'y  +  i  •  x'y)  =  1. 


36  PROPERTIES  OF  DEVELOPED  FUNCTIONS. 

Special  Case. — We  have  the  equalities: 

{ab  -\-  a' b')'  =  ab'  -[■  a  b, 
{ab' -\- a' b)  =  ab -\- a  b' , 

which  may  easily  be  demonstrated  in  many  ways;  for  instance, 
by  observing  that  the  two  sums  {ab -{■  a' b')  and  {ab'+db) 
combined  form  the  development  of  i ;  or  again  by  performing 
the  negation  {ab  -{■  a  b' )'  by  means  of  De  Morgan's  formulas 
(§  25). 

From  these  equalities  we  can  deduce  the  following  equality: 
{ab' ■\-  a  b  =  o)  =  {ab  +  a  b'  =  i), 

which  result  might  also  have  been  obtained  in  another  way 
by  observing  that  (§  18) 

(a  =  /5)  =  {ab'-\-  ab  =  o)  =  [{a  +  b')  (a'+  ^)  =  i], 

and  by  performing  the  multiplication  indicated  in  the  last 
equality. 

.  Theorem. —  We  have  the  following  equivalences:^ 

{a  =  bc'+  b'c)  =  (b  =  ac'+  dc)  =  (c  =  ab'  -{■  a'b). 

For,  reducing  the  first  of  these  equalities  so  that  its  second 
member  will  be  o, 

a{bc  +  b' c)  +  a  {be  +  b'c)  =  o', 

abc  +  ab' c  -\-  a  be  +  a  b' c  =  o. 

Now  it  is  clear  that  the  first  member  of  this  equality  is 
symmetrical  with  respect  to  the  three  terms  a,  b,  c.  We  may 
therefore  conclude  that,  if  the  two  other  equalities  which  differ 
from  the  first  only  in  the  permutation  of  these  three  letters 
be  similarly  transformed,  the  same  result  will  be  obtained, 
which  proves  the  proposed  equivalence. 

Corollary.— li  vft  have  at  the  same  time  the  three  inclusions: 
a<ibc'  -\-  b'c  ^     b  <^ac'  -^  dc ,     c<Cab'-\-db, 

we  have  also  the  converse  inclusions,  and  therefore  the 
corresponding  equalities 

a  =  be -\- b'c,     b  =  ac'-\-dc,     c  =  ab' -\-db. 
I  W.  Stanley  Jevons,  Pure  Logic,  1864,  p.  61. 


LIMITS    OF    A    FUNCTION.  37 

For  if  we  transform  the  given  inclusions  into  equalities,  we 
shall  have 

abc -\- ab' c  '^  o,     abc  ■{■  a  be  =  o,     abc -\- a  b' c  =  o, 
whence,  by  combining  them  into  a  single  equality, 
a  be  +  a3V+  a  be  ■\-  a  b'c  =  o. 

Now  this  equahty,  as  we  see,  is  equivalent  to  any  one  of 
the  three  equalities  to  be  demonstrated. 

28.  The  Limits  of  a  Function. — A  term  x  is  said  to  be 
eomprised  between  two  given  terms,  a  and  ^,  when  it  contains 
one  and  is  contained  in  the  other;  that  is  to  say,  if  we  have, 
for  instance, 

a  <^x,     X  <^b, 

which  we  may  write  more  briefly  as 
a  <^x  <ab. 

Such  a  formula  is  called  a  double  inclusion.  When  the 
term  x  is  variable  and  always  comprised  between  two 
constant  terms  a  and  b,  these  terms  are  called  the  limits 
of  X.  The  first  (contained  in  x)  is  called  inferior  limit;  the 
second  (which  contains  x)  is  called  the  superior  limit. 

Theorem. — A  developed  function  is  comprised  between  the  sum 
and  the  product  of  its  coefficients. 

We  shall  first  demonstrate  this  theorem  for  a  function  of 
one  variable, 

ax  +  bx . 

We  have,  on  the  one  hand, 

{ab  <C.a)  <Ci  (abx  <i  ax), 
{ab  <,b)<^{abx  <^bx). 
Therefore 


or 


abx  +  abx  <C  ax  +  bx', 

ab  <C  ax  +  bx'. 
On  the  other  hand, 

(a  <C  a  +  b)  <^  [ax  <^  (a  +  b)x] , 
{b<a+-bX  [bx<  (a  +  b)x']. 


38  FORMULA    OF    PORETSKY. 

Therefore 


or 


ax  +  bx'<^  {a  -{■  b)  {x  •\-  x'), 
ax  -^ bx  <Z.a  -\-  b. 


To  sum  up, 


ab<Z.(ix-'r  bx'<^a  +  b.  Q.  E.  d. 

Remark  i.  This  double  inclusion  may  be  expressed  in  the 
following  form :  * 

For 

f{a)  =  aa  ■\-  ba'  =  a  ■\-  b, 

f{b)  =  ab-\-  bb' ■=ab. 

But  this  form,  pertaining  as  it  does  to  an  equation  of  one 
unknown  quantity,  does  not  appear  susceptible  of  generalization, 
whereas  the  other  one  does  so  appear,  for  it  is  readily  seen 
that  the  former  demonstration  is  of  general  application. 
Whatever  the  number  of  variables  n  (and  consequently  the 
number  of  constituents  2")  it  may  be  demonstrated  in  exactly 
the  same  manner  that  the  function  contains  the  product  of 
its  coefficients  and  is  contained  in  their  sum.  Hence  the 
theorem  is  of  general  application. 

Remark  2. — This  theorem  assumes  that  all  the  constituents 
appear  in  the  development,  consequently  those  that  are  wanting 
must  really  be  present  with  the  coefficient  o.  In  this  case, 
the  product  of  all  the  coefficients  is  evidently  o.  Likewise 
when  one  coefficient  has  the  value  i,  the  sum  of  all  the 
coefficients  is  equal  to   i. 

It  will  be  shown  later  (S  38)  that  a  function  may  reach 
both  its  limits,  and  consequently  that  they  are  its  extreme 
values.  As  yet,  however,  we  know  only  that  it  is  always 
comprised  between  them. 

29.  Formula  of   Poretsky.^ — We    have   the    equivalence 
{x  =  ax  +  bx)  =  {b<Cx<Ca). 

»  EuGEN  MULLER,  Aus  der  Algebra  der  Logik,  Art.  II. 
*  PoRETSKY,    "Sur  les  m^tliodes  pour  resoudre  les  egalit&  logiques". 
{Bull,  de  la  Soc.  phys.-math.  de  Kazan,  Vol.  II,   1884). 


Schroder's  theorem.  39 

Demonstration. — First  multiplying  by  x  both  members  of 
the  given  equality  [which  is  the  first  member  of  the  entire 
secondary  equality],  we  have 

X  =  ax, 

which,  as  we  know,  is  equivalent  to  the  inclusion 

x<^a. 

Now  multiplying  both  members  by  x\  we  have 
o  =  bx, 
which,  as  we  know,  is  equivalent  to  the  inclusion 

b<x. 
Summing  up,  we  have 

{x  ==  ax  ■\-  bx)  <C.(b<Cx<Za). 

Conversely, 

(b  <^  X  <^  a)  <^  (x  =  ax  +  bx') . 

For 

(x  <^  a)  =  (x  =  ax) , 

(b<x)  ^(bx'=o). 

Adding  these  two  equalities  member  to  member  [the  second 
members  of  the  two  larger  equalities], 

(x  ==  ax)  (o  =  bx)J<^  (x  =  ax  +  bx'). 
Therefore 

(b  <^  X  <C.  a)  <C  (x  =  ax  -\-  bx') , 

and  thus  the  equivalence  is  proved. 

30.  Schroder's  Theorem.' — The  equality 

ax  +  bx'  =  o 
signifies  that  x  lies  between  a'  and  b. 
Demonstration  : 

{ax  +  bx'  =  o)  =  {ax  =  o)  {bx  ==  o), 
{ax  ■=  o)  =  {x<C^a'), 
{bx'  =  o)  =  (b<^x). 

»  Schr5der,  Operationskreis  des  Logikkalkuls  (1877),  Theorem  20. 


40  Schroder's  theorem. 

Hence 

{ax  +  bx  =  o)  ==  {b  <C  X  <^  a) . 

Comparing  this  theorem  with  the  formula  of  Poretsky,  we 
obtain  at  once  the  equality 

{ax  +  bx  =  o)  =  {x  =  ax  +  bx) , 

which    may    be    directly  proved  by  reducing   the  formula  of 
Poretsky   to   an   equality  whose  second  member  is  o,  thus: 

{x  =  ax  +  bx)  =  \x{ax  +  b' x:)  +  x  {a  x  +  bx')  =  o] 
=  {ax-\-  bx  =  o). 

If  we  consider  the  given  equality  as  an  equation  in  which 
X  is  the  unknown  quantity,  Poretsky's  formula  will  be  its 
solution. 

From  the  double  inclusion 

b  <Cx<^a' 
we  conclude,  by  the  principle  of  the  syllogism,  that 

b<d. 

This  is  a  consequence  of  the  given  equality  and  is  in- 
dependent of  the  unknown  quantity  x.  It  is  called  the 
resultant  of  the  elimination  of  x  in  the  given  equation.  It  is 
equivalent  to  the  equality 

ab  =  o. 
Therefore  we  have  the  implication 

{ax-^  bx  =  o)  <[  {ab  =  o). 

Taking  this  consequence  into  consideration,  the  solution 
may  be  simplified,  for 

{ab  =  o)  =  {b  =  a'b). 
Therefore 

x  =  a'x  +  bx'  =  ax  +  a  bx 
=  a  bx  +  a  b'  X  +  a  bx  =  ab  +  a'b'x 
=  b  +  a'b'x  =  b  +  a'x. 

This  form  of  the  solution  conforms  most  closely  to  common 
sense:  since  x'  contains  b  and  is  contained  in  a,  it  is  natural 
that  X   should   be   equal  to   the  sum  of  b  and  a  part  of  a' 


RESULTANT    OF    ELIMINATION.  41 

(that  is  to  say,  the  part  common  to  a  and  x).  The  solution 
is  generally  indeterminate  (between  the  limits  a  and  b)',  it  is 
determinate  only  when  the  limits  are  equal, 

«'==  b  f 
for  then 

AT  =  ^  +  a  X  =  b-\-bx=^b  =  d. 

Then  the  equation  assumes  the  form 

{ax  +  ax  =  o)  =  {a  =  x) 

and  is  equivalent  to  the  double  inclusion 

{a  <^x<i_  a)  =  {x  =  a) . 

31.  The  Resultant  of  Elimination. — When  ab  is  not 
zero,  the  equation  is  impossible  (always  false),  because  it  has 
a  false  consequence.  It  is  for  this  reason  that  Schroder 
considers  the  resultant  of  the  elimination  as  a  condition  of 
the  equation.  But  we  must  not  be  misled  by  this  equivocal 
word.  The  resultant  of  the  elimination  of  x  is  not  a  cause  of 
the  equation,  it  is  a  consequence  of  it;  it  is  not  a  sufficient  but 
a  necessary  condition. 

The  same  conclusion  may  be  reached  by  observing  that 
ab  is  the  inferior  limit  of  the  function  ax-\-bx\  and  that 
consequently  the  function  can  not  vanish  unless  this  Hmit  is  o. 

{ab  <^ax-\-  bx)  {ax  +  bx  =  o)  <C  {ab  ==  o). 

We  can  express  the  resultant  of  elimination  in  other  equiv- 
alent forms;  for  instance,  if  we  write  the  equation  in  the  form 

{a  +  x)  {b-V  X)  =  o, 
we  observe  that  the  resultant 

ab  =  o 

is  obtained  simply  by  dropping  the  unknown  quantity  (by 
suppressing  the  terms  x  and  x).  Again  the  equation  may  be 
written: 

a'  X  +  b'  X  =  I 

and  the  resultant  of  elimination: 

a  -\-  b'  =  I . 


42  RESULTANT    OF    ELIMINATION. 

Here  again  it  is  obtained  simply  by  dropping  the  unknown 
quantity.^ 

Remark.     If  in  the  equation 

ax  +  bx  =  o 

we  substitute  for  the  unknown  quantity  x  its  value  derived 
from  the  equations, 

X  =  a' X  +  bx  ^     X  =  ax  +  b' x  , 
we  find 

{abx  +  abx  =  o)  =  {ab  =  o), 

that  is  to  say,  the  resultant  of  the  elimination  of  x  which,  as 
we  have  seen,  is  a  consequence  of  the  equation  itself.  Thus 
we  are  assured  that  the  value  of  x  verifies  this  equation. 
Therefore  we  can,  with  Voigt,  define  the  solution  of  an  equation 
as  that  value  which,  when  substituted  for  x  in  the  equation, 
reduces  it  to  the  resultant  of  the  elimination  of  x. 

Special  Case.—  When  the  equation  contains  a  term  independent 
of  X,  i.  e.,  when  it  is  of  the  form 

ax  -)r  bx'  +  c  =  o 
it  is  equivalent  to 

{a  +  c)x  +  ((5  +  c)x'  =  o, 

and  the  resultant  of  elimination  is 

{a-\-  c)  (b  +  c)  =  ab  +  e  =  o, 

»  This  is  the  method  of  elimination  of  Mrs.  Ladd-Franklin  and 
Mr.  Mitchell,  but  this  rule  is  deceptive  in  its  apparent  simplicity,  for  it 
cannot  be  applied  to  the  same  equation  when  put  in  either  of  the  forms 

ax  4"  bx'  =  o,         {a  -|-  x')  (l>'-\-  x)  =  I. 

Now,  on  the  other  hand,  as  we  shall  see  (§  54),  for  inequalities  it 
may  be  applied  to  the  forms 

ax  +  bx'^  o,         {a'  -\-  x')  ip'  -j-  ^)  =|=  !• 

and  not  to  the  equivalent  forms 

(a  -}-  x')  (b  ■\-  x)^  o,        ax  ■\-  b'  x'<^  I. 

Consequently,  it  has  not  the  mnemonic  property  attributed  to  it,  for,  to 
use  it  correctly,  it  is  necessary  to  recall  to  which  forms  it  is^  applicable. 


CASE    OF    INDETERMINATION.  43 

whence  we  derive  this  practical  rule:  To  obtain  the  resultant 
of  the  elimination  of  x  in  this  case,  it  is  sufficient  to  equate 
to  zero  the  product  of  the  coefficients  of  x  and  x,  and  add 
to  them  the  term  independent  of  x. 

32.  The  Case  of  Indetermination. — Just  as  the  resultant 

a^  =  o 

corresponds  to  the  case  when  the  equation  is  possible,  so  the 
equality 

a  +  d  =  o 

corresponds  to  the  case  of  absolute  indetermination.  For  in 
this  case  the  equation  both  of  whose  coefficients  are  zero 
(a  =  o),  {b  =  6),  is  reduced  to  an  identity  (0  =  0),  and 
therefore  is  "identically"  verified,  whatever  the  value  of  x  may 
be;  it  does  not  determine  the  value  of  x  at  all,  since  the 
double  inclusion 

b<^x  <i_a 
then  becomes 

o<^<i, 

which  does  not  limit  in  any  way  the  variability  of  x.  In  this 
case  we  say  that  the  equation  is  indeterminate. 

We  shall  reach  the  same  conclusion  if  we  observe  that 
{a  +  b)  is  the  superior  limit  of  the  function  ax  +  bx'  and  that, 
if  this  limit  is  o,  the  function  is  necessarily  zero  for  all 
values  of  x, 

{ax  +  bx'<C  a-\-  b)  (a  +  b  =  o)  <C  (ax  +  bx'  =  o). 

Special  Case. — When  the  equation  contains  a  term  in- 
dependent of  X, 

ax  +  bx'+  c  =  o, 

the  condition  of  absolute  indetermination  takes  the  form 

a  ■\-  b  ■\-  c  =  o. 
For 

ax  +  bx' -\-  c  =  {a  ■\-  c)x  -\-  {b  ■\-  c)x', 
{a^  c)-\-  {b-\-  c)  =  a-\-  b-\-  c=  o. 


44  SUMS    AND    PRODUCTS    OF    FUNCTIONS. 

33.  Sums  and  Products  of  Functions. — It  is  desirable 
at  this  point  to  introduce  a  notation  borrowed  from  mathe- 
matics, which  is  very  useful  in  the  algebra  of  logic.  Let /(a;) 
be  an  expression  containing  one  variable  j  suppose  that  the 
class  of  all  the  possible  values  of  x  is  determined;  then  the 
class  of  all  the  values  which  the  function  /{x)  can  assume 
in  consequence  will   also  be  determined.     Their  sum  will  be 

represented    by  '^/{x)  and  their  product  by  J^A-a;).     This 

X  X 

is  a  new  notation  and  not  a  new  notion,  for  it  is  merely  the 
idea  of  sum  and  product  applied  to  the  values  of  a  function. 

When  the  symbols  ^  ^^^^  O  ^'^^  applied  to  propositions, 
they  assume  an  interesting  significance; 

Y\{f{x)  =  o]       ' 

X 

means  that  f{x)  =  o  is  true  for  every  value  of  x\  and 

2[/(^)  =  o] 

X 

that  f{x^  =  o  is  true  for  some  value  of  x.  For,  in  order 
that  a  product  may  be  equal  to  i  (/'.  <f.,  be  true),  all  its  factors 
must  be  equal  to  i  {i.  e.,  be  true);  but,  in  order  that  a  sum 
may  be  equal  to  i  (/.  <?.,  be  true),  it  is  sufficient  that  only 
one  of  its  summands  be  equal  to  i  {i.  e.,  be  true).  Thus  we 
have  a  means  of  expressing  universal  and  particular  propositions 
when  they  are  applied  to  variables,  especially  those  in  the 
form:  "For  every  value  of  x  such  and  such  a  proposition  is 
true",  and  "For  some  value  of  x,  such  and  such  a  proposition 
is  true",  etc. 

For  instance,  the  equivalence 

(a  =  d)  =  {ac  =  be)  {a  -V  c  =  b  -^  c) 

is  somewhat  paradoxical  because  the  second  member  contains 
a  term  (c)  which  does  not  appear  in  the  first.  This  equivalence 
is  independent  of  c,  so  that  we  can  write  it  as  follows, 
considering  r  as  a  variable  x 

Y\[i^  =  b)  =  {ax  =  bx)  (a  +  X  =  b  +  x)], 


SUMS   AND    PRODUCTS    OF    FUNCTIONS.  45 

or,  the  first  member  being  independent  of  x, 

{a  =-  b)  =  Yi^^'^^  =  bx)  {a-{-  x  =  b-\-  x)]. 

X 

In  general,  when  a  proposition  contains  a  variable  term, 
great  care  is  necessary  to  distinguish  the  case  in  which  it  is 
true  for  ei-ery  value  of  the  variable,  from  the  case  in  which 
it   is    true  only  for  some  value  of  the  variable.'     This   is  the 

purpose  that  the  symbols  J~[  and  ^  serve. 

Thus  when  we  say  for  instance  that  the  equation 

ax-V  bx  =  o 

is  possible,  we  are   stating  that  it  can  be   verified  by  some 
value  of  x;  that  is  to  say, 

2(ax  +  bx'  ==  o), 

'  X 

and,  "since    the  necessary  and  sufficient  condition  for  this  is 
that  the  resultant  {ab  =  o)  is  true,  we  must  write 

^{ax-\-  bx'  =  o)  =  (ab  =  o), 

X 

although  we  have  only  the  implication 

(ax  +  bx'  =  o)  <C  (ab  ==  o) . 

On  the  other  hand,  the  necessary  and  sufficient  condition 
for  the  equation  to  be  verified  by  every  value  of  x  is  that 

a  +  b  =  o. 

Demonstration. — i.  The  condition  is  sufficient,  for  if 

(aAr  b  =  o)  =  (a  =  6)  (b  =  o), 

we  obviously  have 

ax  +  bx'  =  o 

whatever  the  value  of  x;  that  is  to  say, 
]^(a:»:  +  bx  ==  o). 


I  This  is  the  same  as  the  distinction  made  in  mathematics  between 
identities  and  equations,  except  that  an  equation  may  not  be  verified  by 
any  value  of  the  variable. 


46  INCLUSION    AND    INDETERMINATES. 

2.  The  condition  is  necessary,  for  if 

X 

the  equation  is  true,  in  particular,  for  the  value  x  =  a;  hence 
a-V  b  =  o. 
Therefore  the  equivalence 

]~[(aA:  +  bx'  =  o)  =  (a  +  b  =  o) 

X 

is  proved.^  In  this  instance,  the  equation  reduces  to  an 
identity:  its  first  member  is  "identically"  null. 

34.  The  Expression  of  an  Inclusion  by  Means  of  an 
Indeterminate. — The  foregoing  notation  is  indispensable  in 
almost  every  case  where  variables  or  indeterminates  occur  in 
one  member  of  an  equivalence,  which  are  not  present  in  the 
other.  For  instance,  certain  authors  predicate  the  two  following 
equivalences 

{a<^b)  =  (a  ^  bu)  =  {a^  v  =  b), 

in  which  u,  v  are  two  "indeterminates".  Now,  each  of  the 
two  equalities  has  the  inclusion  {a  <^  b)  as  its  consequence, 
as  we  may  assure  ourselves  by  eliminating  u  and  v  respectively 
from  the  following  equalities: 

1.  [a{b'  +  u)  +  a  bu  =  o]  =  \{ab' \  a  b)u  +  au  =  o]. 
Resultant :  ' 

[(a/^'+  a'b)  a  ^  o]  =  (ab'  =  o)  ==  (a<^b). 

2.  [(a  +  v)b'+  dbv  ==  o]  =  {b' v  +  {ab' -\-  a  b)v  ==  o]. 
Resultant : 

[b'{ab'-\-  a'b)  =  o]  =  {ab'  =  o)  +  (a<Cb). 

But  we  cannot  say,  conversely,  that  the  inclusion  implies 
the  two  equahties  for  any  values  of  u  and  v;  and,  in  fact,  we 
restrict  ourselves  to  the  proof  that  this  implication  holds  for 
some  value  of  u  and  v,  namely  for  the  particular  values 


I   EUGEN   MULLER,  Op.   cit. 


INCLUSION    AND    INDETERMINATES.  47 

u  ==  a,     b  =  v; 
for  we  have 

{a  =  ab)  =  {a  <^  b)  =  {a  +  b  =  b). 

But  we  cannot  conclude,  from  the  fact  that  the  implication 
(and  therefore  also  the  equivalence)  is  true  for  some  value  of 
the  indeterminates,  that  it  is  true  for  all;  in  particular,  it  is 
not  true  for  the  values 

for  then  (a  =  bu)    and   (a  +  v  ^  b)   become  (a  =  b),    which 
obviously  asserts  more  than  the  given  inclusion  (a  <C  b).^ 
Therefore  we  can  write  only  the  equivalences 
{a<b)==  ^{a  =  bu)  =  ^{a^v  =  b), 

but  the  three  expressions 

U  V 

are  not  equivalent.* 

1  Likewise  if  we  make 

«  =  o,         z/  =  1 , 
we  obtain  the  equalities 

which  assert  still  more  than  the  given  inclusion. 

2  According  to  the  remark  in  the  preceding  note,  it  is  clear  that 
we  have 

'^^   since   the  equalities  affected  by  the  sign   I   I    may   be    likewise    verified 
by  the  values 

« =  o,         « =  I       and       v  =  o,         v==  l. 

If  we   wish  to   know   within  what  limits  the  indeterminates  u  and  v  are 
variable,  it  is  sufficient  to  solve  with  respect  to  them  the  equations 

{a<Ci>)  =  (a  =  du),         (a  <  <5)  =  (a  +  z/ = /J), 
or 

ab'  =  abu  ^  ab'  -\-  au  ,  a b'=  ab'  +  b'v  -f-  a'b  v', 

or 

a'bu-\-abu' ^o,         a  b'v  -J-  abv'=o. 


48  DOUBLE   INCLUSION   AND    INDETERMINATES. 

35.  The  Expression  of  a  Double  Inclusion  by  Means 
of  an  Indeterminate. — Theorem.     The  dottble  inclusion 

is   equivalent  to    the  equality    x  =  au  -\-  bu    together  with  the 
condition  {b<ia)y  u  being  a  term  absolutely  indeterminate. 

Demonstration, — Let  us    develop    the   equality  in  question, 
x{a'u  +  b'u)  +  X  {au  +  bu)  =  o, 
{ax  +  ax')u  +  {b'  X  +  bx)u  =  o. 
Eliminating  u  from  it, 

a'  b'  X  +  abx  =  o. 
This  equality  is  equivalent  to  the  double  inclusion 

ab<i_x<^a  ■\-  b. 
But,  by  hypothesis,  we  have 

{b  <^a)  =  {ab  ^  b)  =  {a  -^  b  =  a). 
The  double  inclusion  is  therefore  reduced  to 

b<i.x<^a. 
So,  whatever  the  value  of  u,  the  equality  under  consideration 
involves   the    double    inclusion.     Conversely,    the    double   in- 
clusion involves  the  equality,  whatever  the  value  of  x  may  be, 
for  it  is  equivalent  to 

«  a:  +  bx  =0, 
and  then  the  equality  is  simplified  and  reduced  to 
ax  u  +  b' xu  =  o. 

from   which   (by  a   formula  to   be  demonstrated  later  on)  we  derive  the 
solutions 

u  =  ab  •\-  10  [a  •\-  b'),         v  =  ah  -f-  «;  (fl  -f-  b), 
or  simply 

u  =  ab  -\-  wb',  2^=  a  b  -\-wa, 
w  being  absolutely  indeterminate.  We  would  arrive  at  these  solutions 
simply  by  asking:  By  what  term  must  we  multiply  b  in  order  to  obtain 
a?  By  a  term  which  contains  ab  plus  any  part  of  b'.  What  term  must 
we  add  to  a  in  order  to  obtain  b}  A  term  which  contains  ab  plus 
any  part  of  a.  In  short,  u  can  vary  between  ab  and  a-\-b',  v  between 
ab  and  a  ■\-  b. 


DOUBLE    INCLUSIOX    AND    IN  DETERMINATES.  49 

We  can  always  derive  from  this  the  value  of  u  in  terms 
of  X,  for  the  resultant  (ab'xx  =  o)  is  identically  verified. 
The  solution  is  given  by  the  double  inclusion 

b' x<C  u  <^  a  -\-  X. 

Remark. — There  is  no  contradiction  between  this  result, 
which  shows  that  the  value  of  u  lies  between  certain  limits, 
and  the  previous  assertion  that  u  is  absolutely  indeterminate; 
for  the  latter  assumes  that  x  is  any  value  that  will  verify  the 
double  inclusion,  while  when  we  evaluate  u  in  terms  of  x  the 
value  of  X  is  supposed  to  be  determinate,  and  it  is  with 
respect  to  this  particular  value  of  x  that  the  value  of  u  is 
subjected  to  limits.* 

In  order  that  the  value  of  u  should  be  completely  deter- 
mined, it  is  necessary  and  sufficient  that  we  should  have 

b'  X  =  a  +  X, 
that  is  to  say, 

b' xax'  +  (b  +  x)  {a  -\r  :«;)■=  o 
or 

bx  +  ax  =  o. 

Now,  by  hypothesis,  we  already  have 
a  X  +  bx'  =  0. 

If  we  combine  these  two  equalities,  we  find 

(a  +  ^  =  o)  =  (a  =  i)  {b  =  o). 

This  is  the  case  when  the  value  of  x  is  absolutely  in- 
determinate, since  it  lies  between  the  limits  o  and  i. 

In  this  case  we  have  , 

u  ==  b'x  =  a  +  X  =  X. 

In  order  that  the  value  of  u  be  absolutely  indeterminate, 
it  is  necessary  and  sufficient  that  we  have  at  the  same  time 


I  Moreover,  if  we  substitute  for  x  its  inferior  limit  d  in  the  inferior 
limit  of  u,  this  limit  becomes  dd'  =  o ;  and,  if  we  substitute  for  x  its 
superior  limit  a  in  the  superior  limit  of  u,  this  limit  becomes  a  +  a  =1. 

4 


50  EQUATION    INVOLVING   ONE    UNKNOWN. 

b'  X  =^  o,     a  -\-  X  ■=  1, 
or 

b'  X  +  ax  =  o, 
that  is 

a<Cx<^b. 

Now  we  already  have,  by  hypothesis, 

b  <Zx<^a; 
so  we  may  infer 

b  =  X  =  a. 

This    is    the  case    in  which    the  value  of  x  is  completely 
determinate. 

36.  Solution  of  an  Equation  Involving  One  Unknown 
Quantity. — The  solution  of  the  equation 

ax  +  bx'  =  o 
may  be  expressed  in  the  form 

X  =  a  u  +  bu , 

u  being  an  indeterminate,  on  condition  that  the  resultant  of 
the  equation  be  verified;  for  we  can  prove  that  this  equality 
implies  the  equality 

ab' X  +  a  bx  =  o, 

which  is  equivalent  to  the  double  inclusion 

a  b  <^x  <^a  -\-  b. ' 

Now,  by  hypothesis,  we  have 

{ab  =  o)  =  {a'b  =  b)  =^  (a'+  b  =  a). 

Therefore,  in  this  hypothesis,  the  proposed  solution  implies 
the  double  inclusion 

b  <^x<^d  \ 
which  is  equivalent  to  the  given  equation. 

Remark. — In  the  same  hypothesis  in  which  we  have 
{ab=^o)^{b<d), 

we  can  always  put  this  solution  in  the  simpler  but  less  sym- 
metrical forms 

x  =  b-\-du^     X  =  d(b  +  u). 


EQUATION    INVOLVING    ONE    UNKNOWN.  5 1 

For 

1.  We  have  identically 

b  ==  bu  -\r  bu. 
Now 

{b<^a)<^{bu<^a'u). 
Therefore 

{x  =  bu  +  du)  =  {x  ='  b  -{-  a  u). 

2.  Let  us  now  demonstrate  the  formula 

X  =  a  b  +  a  u . 
Now 

a  b  =  b. 

Therefore 

X  =  b  -V  du 

which  may  be  reduced  to  the  preceding  form. 
Again,  we  can 'put  the  same  solution  in  the  form 
X  =  d  b  -{■  u{ab  +  d  b')^ 
which  follows  from  the  equation  put  in  the  form 

ab' x-V  d bx  =  o, 
if  we  note  that 

d -\-  b  =  ab  \  d b  +  d b' 
-and  that 

ud  b  <^d  b. 

This  last  form  is  needlessly  compHcated,  since,  by  hypothesis, 

ab  =  o. 

Therefore  there  remains 

x  =  d  b  +  ud  b' 

which  again  is  equivalent  to 

x  =  b  +  ud, 
since 

d  b  =  b     and     d  =  d  b  +  d  b'. 

Whatever  form  we  give  to  the  solution,  the  parameter  u 
in  it  is  absolutely  indeterminate,  /.  <?.,  it  can  receive  all  possible 
values,  including  o  and  i;  for  when  «  ^  o  we  have 

4* 


52  EQUATION    INVOLVING    ONE   UNKNOWN. 

X  =  b, 
and  when  u=  \  we  have 

X  ==  a , 
and  these  are  the  two  extreme  values  of  x. 

Now  we  understand  that  x  is  determinate  in  the  particular 
case  in  which  a  =  b,  and  that,  on  the  other  hand,  it  is 
absolutely  indeterminate  when 

b  =  o,     a  ^  I,     (or  a  =  o). 

Summing  up,  the  formula 

X  =  a'u  +  bu 

replaces  the  "limited"  variable  x  (lying  between  the  limits  a 
and  b)    by  the  "unlimited"  variable  u  which  can  receive  all 
possible  values,  including  o  and  i. 

Remark^ — The  formula  of  solution 
x  ==  a  X  +  bx 

is    indeed    equivalent  to   the   given  equation,  but  not  so  the 
formula  of  solution 

X  =  a  u  +  bu 
as  a  function  of  the  indeterminate  u.    For  if  we  develop  the 
latter  we  find 

ab' x-\-  a  bx' ■\-  abixu  +  x  u)  +  a  b' {xu  ■\-  x'u)  =  o, 
and  if  we  compare  it  with  the  developed  equation 

ab  +  ab'  X  +  a'  bx'  =  o, 
we  ascertain  that  it  contains,  besides  the  solution,  the  equality 

ab{xu' ■\-  x'u)  =  o, 
and  lacks  of  the  same  solution  the  equality 

a  b'  {xu  -{■  x'  u)  =  o. 
Moreover  these  two  terms  disappear  if  we  make 
u  =  X 
and  this  reduces  the  formula  to 

X  =  a  X  +  bx'. 

I  PORETSKY.     Sept  his.  Chaps.  XXXIII  and  XXXIV. 


ELIMINATION    OF    SEVERAL   UNKNOWNS,  -53 

From  this  remark,  Poretsky  concluded  that,  in  general,  the 
solution  of  an  equation  is  neither  a  consequence  nor  a  cause 
of  the  equation.  It  is  a  cause  of  it  in  the  particular  case  in  which 

ab  =  o, 

and  it  is  a  consequence  of  it  in  the  particular  case  in  which 

{ab' ^  o)  =  (^  +  ^  =  i). 

But  if  ab  is  not  equal  to  o,  the  equation  is  unsolvable  and 
the  formula  of  solution  absurd,  which  fact  explains  the 
preceding  paradox.     If  we  have  at  the  same  time 

ab  =  o     and     a-{-b  =  i., 

the  solution  is  both  consequence  and  cause  at  the  same  time, 
that  is  to  say,  it  is  equivalent  to  the  equation.  For  when 
a  =  b  the  equation  is  determinate  and  has  only  the  one 
solution 

X  =  a'  =  b. 

Thus,  whenever  an  equation  is  solvable,  its  solution  is  one 
of  its  causes;  and,  in  fact,  the  problem  consists  in  finding  a 
value   of  X  which  will  verify  it,  /.  <?.,  which  is  a  cause  of  it. 

To  sum  up,  we  have  the  following  equivalence: 

(ax  +  bx'  =  o)  =  (ab  ==  o)^(^  =  a'u  +  bu) 

which  includes  the  following  implications: 

{ax  +  bx  ^  o)  <^  {ab  =  o), 
{ax  +  bx'  =  o)  <^  2^"^  "^  ^'^  "*"  ^"')» 

u 

{ab  =  oy^{x  =  a'u  +  bu')  <^  {ax  +  bx  =  o). 

u 

37.    Elimination  of  Several  Unknown   Quantities. — 

We  shall  now  consider  an  equation  involving  several  unknown 
quantities  and  suppose  it  reduced  to  the  normal  form,  i.  <?., 
its  first  member  developed  with  respect  to  the  unknown 
quantities,  and  its  second  member  zero.  Let  us  first  concern 
ourselves  with  the  problem  of  elimination.  We  can  ehminate 
the  unknown  quantities  either  one  by  one  or  all  at  once. 


54  ELIMINATION    OF   SEVERAL    UNKNOWNS. 

For  instance,  let 
(i)       (f){x,y,  z)  =  axyz  +  bxyz' ^  cxy  z  +  dxy  z' 

Arfx'yz  +  gx'yz  +  hx'y'z  +  kx'y'z'  =  o 
be  an  equation  involving  three  unknown  quantities. 

We  can  eliminate  z  by  considering  it  as  the  only  unknown 
quantity,  and  we  obtain  as  resultant 

{axy  +  cxy' -\rfxy  +  hx y)  {bxy  +  dxy  +  gx y  +  kx y  )  =  o 
or 

(2)  abxy  ■\-  cdxy  ^fgxy-^-hkxy  =0. 

If  equation  (i)  is  possible,  equation  (2)  is  possible  as  well; 
that  is,  it  is  verified  by  some  values  of  x  and  y.  Accordingly 
we  can  eliminate  y  from  the  equation  by  considering  it  as 
the  only  unknown  quantity,  and  we  obtain  as  resultant 

{abx  -Vfgx  )  {cdx  +  hkx  )  =  o 
or 

(3)  abcdx  -^-fghkx  =  o. 

If  equation  (i)  is  possible,  equation  (3)  is  also  possible; 
that  is,  it  is  verified  by  some  values  of  x.  Hence  we  can 
eliminate  x  from  it  and  obtain  as  the  final  resultant, 

abed  .fghk  =  o 

which  is  a  consequence  of  (i),  independent  of  the  unknown 
quantities.  It  is  evident,  by  the  principle  of  symmetry,  that 
the  same  resultant  would  be  obtained  if  we  were  to  eliminate 
the  unknown  quantities  in  a  different  order.  Moreover  this 
result  might  have  been  foreseen,  for  since  we  have  (§  28) 

abcdfghk  <^  (p  (x,  y,  z) , 

<p{x,  y,  z)  can  vanish  only  if  the  product  of  its  coefficients 
is  zero: 

[Vix,y,  2;)  =  o]  <  {abcdfghk  =  o). 

Hence  we  can  eliminate  all  the  unknown  quantities  at  once 
by  equating  to  o  the  product  of  the  coefficients  of  the 
function  developed  with  respect  to  all  these  unknown  quantities. 

We  can  also  eliminate  some  only  of  the  unknown  quantities 
at  one  time.     To  do  this,  it  is  sufficient  to  develop  the  first 


VALUES    OF   A    FUNCTION.  5§ 

member  with  respect  to  these  unknown  quantities  and  to 
equate  the  product  of  the  coefficients  of  this  development 
to  o.  This  product  will  generally  contain  the  other  unknown 
quantities.  Thus  the  resultant  of  the  elimination  of  z  alone, 
as  we  have  seen,  is 

abxy  +  cdxy  +/gx  y  +  hkx  y  =  o 

and  the  resultant  of  the  elimination  of  y  and  z  is 
abcdx -\- fghkx  =  o. 

These  partial  resultants  can  be  obtained  by  means  of  the 
following  practical  rule:  Form  the  constituents  relating  to  the 
unknown  quantities  to  be  retained;  give  each  of  them,  for  a 
coefficient,  the  product  of  the  coefficients  of  the  constituents 
of  the  general  development  of  which  it  is  a  factor,  and  equate 
the  sum  to  o. 

38.  Theorem  Concerning  the  Values  of  a  Function: — 

All  the  values  which  can  be  assumed  by  a  function  of  any  number 
of  variables  f{x,y,  z  .  .  .)  are  given  by  the  formula 

abc  ...k-\-u{a  +  b-{-c^...  +  k), 

in  which  u  is  absolutely  indeterminate,  and  a,  b-  c  .  .  .,  k  are 
the  coefficients  of  the  development  of  f 

Demonstration. — It  is  sufficient  to  prove  that  in  the  equality 

f{x,  y,  z  ..  )  =  abc  ...k-\-u{a-\-b-\-c-\-...-{-k) 

u  can  assume  all  possible  values,  that  is  to  say,  that  this 
equality,  considered  as  an  equation  in  terms  of  u,  is  in- 
determinate. 

In  the  first  place,  for  the  sake  of  greater  homogeneity,  we 
may  put  the  second  member  in  the  form 

u  abc  . ..  k-\-  u(a  -^  b  +  c  ^  . .  .  +  ^), 
for 

abc  . .  ,  k  =  uabc  .  .  .k  ■\-  u  abc  . . .  k, 
and 

uabc  . . .  k<^  u(a  +  b  +  c  +  . . .  +  k). 

Reducing  the  second  member  to  o  (assuming  there  are 
only  three  variables  x,  y,  z) 


56  VALUES    OF    A    FUNCTION. 

(axyz  +  bxy  z  +  cxy  z  +  . . .  +  kx  y  z) 

x[ua  d' c  . .  .^'  +  u  (a  +  d'  +  c  +  . . .  +  Ji:)] 
+  (a  xyz  +  b  xyz  4-  c  xy  z  ■\- .  ..■\-  k  x  y  z  ) 
rx\u{a  +  ^  +  ^  +  ...  +  ^)  +  ti  abc  .../§]  =  o, 
or  more  simply 

u{a-\-  b  -\-  c  ■\- . .  .-\-  k){a  xyz  +  b  xyz  -^^  c  xy  z-\-  ...-^k  xyz) 
+  u  (a  +b  +c  + .  ..  +  ^  )(axyz  +  bxyz  +  ...  +  ^xy  z  )  =  o. 

If  we  eliminate  all  the  variables  x,  y,  z,  but  not  the  in- 
determinate u,  we  get  the  resultant 

u{a  +  b  +  c  +  . . .  +  k)a  b' c  . .  .k 

■\-  u  {a  -\-  b  -\-  c  -\- .  .  .-\-  k  )abc  .  .  .k  =  o. 

Now  the  two  coefficients  of  u  and  u  are  identically  zero; 
it  follows  that  u  is  absolutely  indeterminate,  which  was  to  be 
proved.^ 

From  this  theorem  follows  the  very  important  consequence 
that  a  function  of  any  number  of  variables  can  be  changed 
into  a  function  of  a  single  variable  without  diminishing  or 
altering  its  "variability". 

Corollary. — A  function  of  any  number  of  variables  can 
become  equal  to  either  of  its  limits. 

For,  if  this  function  is  expressed  in  the  equivalent  form 

abc .  .  .k-\-  u{a-\-  b  -^r  c  ^r  '•  '■\-  k), 

it  will  be  equal  to  its  minimum  {abc . .  .k)  when  u  ==  o,  and 
to  its  maximum  (a  +  b  +  c  +  . . .  +  k)  when  u  =  i. 

Moreover  we  can  verify  this  proposition  on  the  primitive 
form  of  the  function  by  giving  suitable  values  to  the 
variables. 

Thus  a  function  can  assume  all  values  comprised  between 
its  two  limits,  including  the  limits  themselves.  Consequently, 
it  is  absolutely  indeterminate  when 

abc  . . .  k  =  o    and    a  +  b  -\-  c  -Y  . .  .-\r  k  ^i 
at  the  same  time,  or 

abc  ...k=o  =  abc...k. 

I  Whitehead,  Universal  Algebra,  Vol.  I,  S  Zl  (4)' 


IMPOSSIBILITY    AND    INDETERMINATION.  57 

39.  Conditions  of  Impossibility  and  Indetermination.- - 
The  preceding  theorem  enables  us  to  find  the  conditions 
under  which  an  equation  of  several  unknown  quantities  is 
impossible  or  indeterminate.  Let  fix,  y,  z  .  .  .)  be  the  first 
member  supposed  to  be  developed,  and  a,  d,  c  .  .  .,  /^  its 
coefficients.  The  necessary  and  sufficient  condition  for  the 
equation  to  be  possible  is 

adc  . . .  k  =  o. 

For,  (i)  if  y  vanishes  for  some  value  of  the  unknowns, 
its  inferior  limit  abc . . .  k  must  be  zero;  (2)  if  abc. .  .kis  zero, 
/may  become  equal  to  it,  and  therefore  may  vanish  for  certain 
values  of  the  unknowns. 

The  necessary  and  sufficient  condition  for  the  equation  to 
be  indeterminate  (identically  verified)  is 

a-\-b-^c...-\-k  =  o. 

For,  (i)  if  a-\-b-¥c-\-...-\-k  is  zero,  since  it  is  the 
superior  limit  of  /,  this  function  will  always  and  necessarily 
be  zero;  (2)  if  /  is  zero  for  all  values  of  the  unknowns, 
a  +  b-'r  €■{■...  -V  k  will  be  zero,  for  it  is  one  of  the  values 
of/ 

Simiming  up,  therefore,  we  have  the  two  equivalences 

2  [/(^'  ^'  z,  "  ■)  =  o]  =  iflbc. . .  k  =  o). 

n  [/(^'  y^  2 . . .)  =  o]  =  (^  +  <^  +  r. . .  +  /^  =  o). 

The  equality  abc . . .  k  ■=-=  o  \s,  as  we  know,  the  resultant 
of  the  elimination  of  all  the  unknowns;  it  is  the  consequence 
that  can  be  derived  firom  the  equation  (assumed  to  be  veri- 
fied) independently  of  all  the  imknowns. 

40.  Solution  of  Equations  Containing  Several  Un- 
known Quantities. — On  the  other  hand,  let  us  see  how 
we  can  solve  an  equation  with  respect  to  its  various  un- 
knowns, and,  to  this  end,  we  shall  limit  ourselves  to  the 
case  of  two  unknowns 

axy  ■\-  bxy  +  cxy  +  dx  y  =  o. 


$8      •  EQUATIONS    CONTAINING   SEVERAL    UNKNOWNS. 

First  solving  with  respect  to  x, 

X  =  (ajy  +  b  y)  x  +(cy  +  dy  )  x  . 

The  resultant  of  the  elimination  of  x  is 

acy  +  bdy  =  o. 

If  the  given  equation  is  true,  this  resultant  is  true. 
Now  it  is  an  equation  involving  y  only;  solving  it, 

7  =  («'  +  /)  y  +  bdy  . 

Had  we  eliminated  y  first  and  then  x^  we  would  have 
obtained  the  solution 

y  ==  {ax  +  ex')  y  +  {bx  +  dx) y 
and  the  equation  in  x 

abx  +  cdx  =  o, 
whence  the  solution 

x=  {a  -\-b)  X  ■\-  cdx  . 

We  see  that  the  solution  of  an  equation  involving  two 
unknown  quantities  is  not  symmetrical  with  respect  to  these 
unknowns;  according  to  the  order  in  which  they  were  elim- 
inated, we  have  the  solution 

X  ==  (a'y  +  b'y)  x  +  ify-Vdy)  x  , 
y  =  {a  -\-  c)  y  +  bdy  , 
or  the  solution 

X  ==  {a  ■\-  b')  X  +  cdx( 

y=  (a  X  +  c  x)  y  +  {bx  +  dx)  y  . 

If  we  replace  the  terms  :*:,  y,   in  the  second  members  by 

indeterminates  u,  v,  one  of  the  unknowns  will  depend  on  only 

one  indeterminate,   while  the  other  will  depend  on  two.    We 

shall  have  a  symmetrical  solution  by  combining  the  two  formulas, 

X'=  {a  +  b')  u  +  cdu  , 

y  =  {a  +  c)  V  +  bdv  , 

but  the  two  indeterminates  u  and  v  will  no  longer  be  inde- 
pendent of  each  other.  For  if  we  bring  these  solutions  in- 
to the  given  equation,  it  becomes 


PROBLEM    OF    BOOLE.  5^ 

abed  +  ab' c  uv  +  abd'uv    +  a  cd'u'v  +  b' c  dti  v  ==  o 

or  since,  by  hypothesis,  the  resultant  abed  =  o  is  verified, 

ab' c  uv  +  a'bduv  -\-  a  eduv  +  b'e'du'v   =  o. 

This  is  an  "equation  of  condition"  which  the  indeterminates 
u  and  V  must  verify;  it  can  always  be  verified,  since  its 
resultant  is  identically  true, 

ab  e  .abd.aed.bed=aa  .bb.ee.  dd  =  o, 

but  it  is  not  verified  by  any  pair  of  values  attributed  to  u 
and  V. 

Some  general  symmetrical  solutions,  /.  e.,  symmetrical 
solutions  in  which  the  unknowns  are  expressed  in  terms  of 
several  independent  indeterminates,  can  however  be  found. 
This  problem  has  been  treated  by  •Schroder'',  by  White- 
head ^  and  by  Johnson.  ^ 

This  investigation  has  only  a  purely  technical  interest;  for, 
from  the  practical  point  of  view,  we  either  wish  to  eliminate 
one  or  more  unknown  quantities  (or  even  all),  or  else  we  seek 
to  solve  the  equation  with  respect  to  one  particular  unknown. 
In  the  first  case,  we  develop  the  first  member  with  respect 
to  the  unknowns  to  be  eliminated  and  equate  the  product  of 
its  coefficients  to  o.  In  the  second  case  we  develop  with 
respect  to  the  unknown  that  is  to  be  extricated  and  apply 
the  formula  for  the  solution  of  the  equation  of  one  unknown 
quantity.  If  it  is  desired  to  have  the  solution  in  terms  of 
some  unknown  quantities  or  in  terms  of  the  known  only,  the 
other  unknowns  (or  all  the  unknowns)  must  first  be  eliminated 
before  performing  the  solution. 

41.  The  Problem  of  Boole. — According  to  Boole  the 
most  general  problem  of  the  algebra  of  logic  is  the  follow- 
ing'*: 

1  Algebra  der  Logik,  Vol.  I,  S  24. 

2  Universal  Algebra,  Vol.  I,  S§  35—37- 

^  "Sur  la  th^orie  des  ^galites  logiques",  Bibl.  du  Cong,  intern,  de  Phil., 
Vol.  Ill,  p.  185  (Paris,   1 901). 

4  Laws  of  Thought,  Chap.  IX,  §  8. 


60  PROBLEM   OF   BOOLE. 

Given  any  equation  (which  is  assumed  to  be  possible) 

f{x,  y,  z,...)  =  o, 

and,  on  the  other  hand,  the  expression  of  a  term  /  in  terms 
of  the  variables  contained  in  the  preceding  equation 

/  =  90  ix,y,z,...), 

to  determine  the  expression  of  /  in  terms  of  the  constants 
contained  in  /  and  in  (p. 

Suppose  /  and  (p  developed  with  respect  to  the  variables 
x,y,  z . . .  and  \tt  p^,,  p2,p^, . .  -  be  their  constituents: 

/(^,  J/,  0, . . .)  =  AA  +  B/,  +  C/3  +  . . ., 
(f  {x,y,  0, . . .)  =  ^/i  +  f>p2  +  ^/3  +  . . .. 

Then  reduce    the   equation  which   expresses  /  so   that  its 
second  member  will  be  o: 

{tcp  +  t'  cp  =  o)^  {{a'p,  +  b'p^  +  //,  +  ...)/ 

+  (^A  +  bp2+cp^  +  ...)/'=  o]. 
Combining  the  two   equations  into   a  single  equation  and 
developing  it  with  respect  to  t: 

[{A  +  a)p,  +  (^  +  b')p,  +  (C+  /)/3  +  . . .]  / 
+  [{A  +  a)j>,  +  (^  +  b)p^  +  (C  +  c)p^  +  ...]/=  o. 

This    is    the    equation    which  gives  the  desired  expression 
QJt  /.     Eliminating  t,  we  obtain  the  resultant 

Ap,  +  Bp^  +  C/3  +  . . .  =  o,  \ 

as  we  might  expect.  If,  on  the  other  hand,  we  wish  to 
eliminate  x,y,z,...  (t.  e.,  the  constituents  /i ,  p2,  p^,-  •  •),  we 
put  the  equation  in  the  form 

(A  +  af+  a/)A  +  (^+  b't+  b/)p^+  (C+ci+<:/)p.  +  ...=  o, 
and  the  resultant  will  be 

(A  +  at+a/)  (B  +  b't+b/)  (C+ ct  +  c/) ...^  o, 

an  equation  that  contains  only  the  unknown  quantity  /  and 
the  constants  of  the  problem  (the  coefficients  of  y  and  of  (p). 
From  this  may  be  derived  the  expression  of  /  in  terms  of 
these  constants.    Developing  the  first  member  of  this  equation 

(A  +  a){B+b)(C+c)...Xf+(A  +  a)(B  +  b){C+c)...xt'==o. 


METHOD    OF    PORETSKY.  6l 

The  solution  is 

t^{A-\-a)  (B+d)  (C+c)...  +  u(^a  +  B'd+  C'c  +  ...). 

The  resultant  is  verified  by  hypothesis  since  it  is 

ABC.  ..  =  o, 

which  is  the  resultant  of  the  given  equation 

/  (x,  y,z,...)  =  o. 

We  can  see  how  this  equation   contributes   to    restrict  the 

variability  of  /.    Since  /  was  defined  only  by  the  function  (p, 

it  was  determined  by  the  double  inclusion 

adc...<^f<C,a-\-i-{-c  +  .... 

Now   that  we   take  into  account  the  condition  y==  o,  /  is 

determined  by  the  double  inclusion 

(A  +  a)  (B  +  d)  (C+^)...</<  (A'a  +  B'^  +  C'c+...).' 

The  inferior  limit  can  only  have  increased  and  the  superior 
limit  diminished,  for 

ai>c...<,  (A  +  a)  (B+d)  (C+c)... 
and 

A'a  +  B'd+C'c...<Ca  +  d  +  c 

The  limits  do  not  change  if  A  =  B=^C=..,==o,  that 
is,  if  the  equation  /=  o  is  reduced  to  an  identity,  and  this 
was  evident  a  priori. 

42.  The  Method  of  Poretsky. — The  method  of  Boole 
and  Schroder  which  we  have  heretofore  discussed  is  clearly 
inspired  by  the  example  of  ordinary  algebra,  and  it  is  summed 
up  in  two  processes  analogous  to  those  of  algebra,  namely 
the  solution  of  equations  with  reference  to  unknown  quantities 
and  elimination  of  the  unknowns.  Of  these  processes  the 
second  is  much  the  more  important  from  a  logical  point  of 
view,  and  Boole  was  even  on  the  point  of  considering  de- 
duction  as   essentially  consisting  in  the  elimination  of  middle 


»  Whitehead,  Universal  Algebra,  p.  63. 


62  LAW    OF    FORMS, 

terms.  This  notion,  which  is  too  restricted,  was  suggested 
by  the  example  of  the  syllogism,  in  which  the  conclusion 
results  from  the  elimination  of  the  middle  term,  and  which 
for  a  long  time  was  wrongly  considered  as  the  only  type 
of  mediate  deduction.^ 

However  this  may  be,  Boole  and  Schr6d£r  have  exag- 
gerated the  analogy  between  the  algebra  of  logic  and  ordi- 
nary algebra.  In  logic,  the  distinction  of  known  and  unknown 
terms  is  artificial  and  almost  useless.  All  the  terms  are — in 
principle  at  least — known,  and  it  is  simply  a  question,  certain 
relations  between  them  being  given,  of  deducing  new 
relaticms  (unknown  or  not  explicitly  known)  from  these  known 
relations.  This  is  the  purpose  of  Poretsky's  method  which 
we  shall  now  expound.  It  may  be  summed  up  in  three 
laws,  the  law  of  forms,  the  law  of  consequences  and  the 
law  of  causes. 

43.  The  Law  of  Forms. — This  law  answers  the  following 
problem:  An  equality  being  given,  to  find  for  any  term 
(simple  or  complex)  a  determination  equivalent  to  this  equal- 
ity. In  other  words,  the  question  is  to  find  all  the  forms 
equivalent  to  this  equality,  any  term  at  all  being  given  as 
its  first  member. 

We  know  that  any  equality  can  be  reduced  to  a  form  in 
which  the  second  member  is  o  or  i;  /.  <?.,  to  one  of  the 
two  equivalent  forms 

iV"=o,         iV^=  I. 

The  function  N  is  what  Poretsky  calls  the  logical  zero 
of  the  given  equality;  N'  is  its  logical  whole,'^ 

»  In  fact,  the  fundamental  formula  of  elimination 
{ax  -j-  bx  ^=  o)  <;  {ab  =  o) 
is,  as  we  have  seen,    only  another  form  and  a  consequence  of  the  prin- 
ciple of  the  syllogism 

(<5<x<a')<(^<a'). 
2  They    are    called    "logical"    to    distinguish   them  from  the  identical 
zero  and  whole,  i.  e.,  to  indicate  that  these  two  terms  are  not  equal  to  o 
and  I  respectively  except  by  virtue  of  the  data  of  the  problem. 


LAW    OF    CONSEQUENCES.  6$ 

Let    C/  be  any  term;  then  the  determination  of  U: 

U=N'U-^NU' 

is   equivalent    to   the  proposed    equality;    for  we  know  it  is 
equivalent  to  the  equality 

{NU  +  Nlf  =  o)  =  (iV^=  o). 

Let  us  recall  the  signification  of  the  determination 

U=N'  U^  NU'. 

It  denotes  that  the  term  U  is  contained  in  JSt  and  con- 
tains N.  This  is  easily  understood,  since,  by  hypothesis, 
N  is  equal  to  o  and  N"  to  i.  Therefore  we  can  formulate 
the  law  of  forms  in  the  following  way: 

To  obtain  all  the  forms  equivalent  to  a  given  equality,  it 
is  sufficient  to  express  that  any  term  contains  the  logical  zero 
of  this  equality  and  is  contained  in  its  logical  whole. 

The  number  of  forms  of  a  given  equality  is  unlimited;  for 
any  term  gives  rise  to  a  form,  and  to  a  form  different  from 
the  others,  since  it  has  a  different  first  member.  But  if  we 
are  limited  to  the  universe  of  discourse  determined  by  n 
simple  terms,  the  number  of  forms  becomes  finite  and  de- 
terminate. For,  in  this  limited  universe,  there  are  2«  con- 
stituents. Now,  all  the  terms  in  this  universe  that  can  be 
conceived  and  defined  are  sums  of  some  of  these  con- 
stituents. Their  number  is,  therefore,  equal  to  the  number 
of  combinations  that  can  be  made  with  2«  constituents, 
namely  2^*  (including  o,  the  combination  of  o  constituent, 
and  I,  the  combination  of  all  the  constituents).  This  will 
also  be  the  number  of  different  forms  of  any  equality  in  the 
universe  in  question. 

44.  The  Law  of  Consequences. — We  shall  now  pass  to 
the  law  of  consequences.  Generalizing  the  conception  of 
Boole,  who  made  deduction  consist  in  the  elimination  of 
middle  terms,  Poretsky  makes  it  consist  in  the  elimination 
of  known  terms  {connaissatices).  This  conception  is  explained 
and  justified  as  follows. 


64  LAW    OF    CONSEQUENCES. 

All  problems  in  which  the  data  are  expressed  by  logical 
equalities  or  inclusions  can  be  reduced  to  a  single  logical 
equality  by  means  of  the  formula  ^ 

{A=  o)  (£=  o)  (C=  o)...  =  (A  +  B+  C...  =  o). 

In  this  logical  equality,  which  sums  up  all  the  data  of  the 
problem,  we  develop  the  first  member  with  respect  to  all 
the  simple  terms  which  appear  in  it  (and  not  with  respect 
to  the  unknown  quantities).  Let  n  be  the  number  of  simple 
terms;  then  the  number  of  the  constituents  of  the  develop- 
ment of  I  is  2".  Let  m  (<  2«)  be  the  number  of  those 
constituents  appearing  in  the  first  member  of  the  equality. 
All  possible  consequences  of  this  equaUty  (in  the  universe 
of  the  n  terms  in  question)  may  be  obtained  by  forming  all 
the  additive  combinations  of  these  m  constituents,  and  equat- 
ing them  to  o;  and  this  is  done  in  virtue  of  the  formula 

(A  +  £=o)<{A  =  o). 

We  see  that  we  pass  from  the  equality  to  any  one  of  its 
consequences  by  suppressing  some  of  the  constituents  in  its 
first  member,  which  correspond  to  as  many  elementary  equal- 
ities (having  o  for  second  member),  /.  e.,  as  many  as  there  are 
data  in  the  problem.  This  is  what  is  meant  by  "eliminating 
the  known  terms". 

The  number  of  consequences  that  can  be  derived  from 
an  equality  (in  the  universe  of  n  terms  with  respect  to  which 
it  is  developed)  is  equal  to  the  number  of  additive  com- 
binations that  may  be  formed  with  its  m  constituents;  i.  e., 
2'«.  This  number  includes  the  combination  of  o  constituents, 
which  gives  rise  to  the  identity  0  =  0,  and  the  combination 
of  the  m  constituents,  which  reproduces  the  given  equality. 

Let  us  apply  this  method  to  the  equation  with  one  un- 
known quantity 

ax  +  px  =  o. 


I  We   employ  capitals  to  denote  complex  terms   (logical  functions)  in 
contrast  to  simple  terms  denoted  by  small  letters  (a,  b,  c,  .  .  .) 


LAW    OF    CONSEQUENCES.  6§ 

Developing  it  with  respect  to  the  three  terms  a,  d,  x: 
(a l> X  +  ai>  X  +  a  ^ X  -V  a  bx  =  o) 
=  \ab  ix  -\-  X  )  -\-  ab  X  -\-  a  b X  ==o] 
=  iab  ^  o)   (ab  a;  =  o)  {a  bx  =  o). 

Thus  we  find,  on  the  one  hand,  the  resultant  ab  =  o, 
and,  on  the  other  hand,  two  equalities  which  may  be  trans- 
formed into  the  inclusions 

X  <^  a  +  b,         a  b  <^x. 

But  by  the  resultant  which  is  equivalent  to  b<C.a ,  we  have 
a  +  b  ^  a  ,  a  b  =  b. 

This  consequence  may  therefore  be  reduced  to  the  double 
inclusion 

x<^  a\  b<,x, 

that  is,  to  the  known  solution. 

Let  us  apply  the  same  method  to  the  premises  of  the 
syllogism 

{a<b)  ib<c). 

Reduce  them  to  a  single  equality 

(a<^b)  =  (ab'  =  o),      (b<^c)  =  {bc  =  o),      {a^  +  be  =  o), 

and  seek  all  of  its  consequences. 

Developing  with  respect  to  the  three  terms  a,  b^  c\ 
abc  +  ab  c  -T  ab  c  -\-  a  be  =  o. 

The  consequences  of  this  equality,  which  contains  four 
constituents,  are  i6  (2*)  in  number  as  follows: 

1.  {abc  ^  o)  =  {ab<ic); 

2.  (ab  c  =  o)  =  {ae<^b); 

3.  (all c  =  6)  =  (a<i_b  ^-  c); 

4.  (a'bc  ^  o)  =  (b<i_a  -V  c); 

5.  (abc'  +  al)  c  =  o)  =  (a  <^  be  +  b  c  ) ; 

6.  (abc  +  ab  c  =  o)  =  (ae  =  o)  =  (a<C  c). 


66  LAW    OF    CONSEQUENCES. 

This  is  the  traditional  conclusion  of  the  syllogism.^ 

7.  {abc  +  a  be  =  o)  =  {be  =  o)  =  {b<^c). 
This  is  the  second  premise. 

8.  {all  e  +  ab' c  ^  o)  =  {alJ  =  o)  =  {a<C b). 
This  is  the  first  premise. 

9.  {ab'e  +  a  be  =■  o)  ==  {ae  <ib<^a  Ar  e)\ 

10.  {ab'c' -\- a  be' =  o)  =  {ab' -\-  ab<ie)', 

11.  {abc' ■\-  al?  c  -\-  ab' e  ^0)=  {ai  -\-  ae'  =  o)=^{a<^be)\ 

12.  {abc'  ■\-  ab'e  +  a'/^<r  =  o)  ^=  {ab  c  ■\-  be  ^  o) 

==(a<r<^<0; 

13.  {abc  -\-  ab' c  -\-  abe  =  6)  =  {ae  -^  be  =  6) 

14.  {ab' c  +  ai^V  ■\-  a  be  =  o)  =  (<z3  +  ^  (^^  =0) 

=  (a  <  <^  <  a  +  ^). 

The  last  two  consequences  (15  and  16)  are  those  ob- 
tained by  combining  o  constituent  and  by  combining  all;  the 
first  is  the  identity 

15.  0  =  0, 

which  confirms  the  paradoxical  proposition  that  the  true 
(identity)  is  imphed  by  any  proposition  (is  a  consequence 
of  it);  the  second  is  the  given  equality  itself 

16.  ab  -\-  be  =  o, 

which  is,  in  fact,  its  own  consequence  by  virtue  of  the 
principle  of  identity.  These  two  consequences  may  be  called 
the  "extreme  consequences"  of  the  proposed  equality.  If 
we  wish  to  exclude  them,  we  must  say  that  the  number  of 
the  consequences  properly  so  called  of  an  equality  of  m 
constituents  is  2"' — 2. 


I  It  will  be  observed  that  this  is  the  only  consequence  (except  the 
two  extreme  consequences  [see  the  text  below])  independent  of  b;  there- 
fore it  is  the  resultant  of  the  elimination  of  that  middle  term. 


LAW    OF    CAUSES.  ^ 

45.  The  Law  of  Causes. — The  method  of  finding  the 
consequences  of  a  given  equality  suggests  directly  the  method 
of  finding  its  causes,  namely,  the  propositions  of  which  it  is 
the  consequence.  Since  we  pass  from  the  cause  to  the 
consequence  by  eliminating  known  terras,  ;.  <?.,  by  suppressing 
constituents,  we  will  pass  conversely  from  the  consequence 
to  the  cause  by  adjoining  known  terms,  /.  ^.,  by  adding  con- 
stituents to  the  given  equality.  Now,  the  number  of  con- 
stituents that  may  be  added  to  it,  i.  e.,  that  do  not  already 
appear  in  it,  is  2" — m.  We  will  obtain  all  the  possible 
causes  (in  the  universe  of  the  n  terms  under  consideration) 
by  forming  all  the  additive  combinations  of  these  constituents, 
and  adding  them  to  the  first  member  of  the  equality  in  virtue 
of  the  general  formula 

(^  +  ^  =  o)<  (^  =  o), 

which  means  that  the  equality  {A  =  o)  has  as  its  cause  the 
equality  {A  -{■  £  =  0),  in  which  B  is  any  term.  The  number 
of  causes  thus  obtained  will  be  equal  to  the  number  of  the 
aforesaid  combinations,  or  2^^-m. 

This  method  may  be  applied  to  the  investigation  of  the 
causes  of  the  premises  of  the  syllogism 

{a<b)  {b<c), 

which,  as  we  have  seen,  is  equivalent  to  the  developed 
equality 

abc  +  ab  c  -\-  ab  c  -\-  a  be  =0. 

This  equality  contains  foiir  of  the  eight  (23)  constituents 
of  the  universe  of  three  terms,  the  four  others  being 

/  /    /  t    r   / 

abc,  abc,  abc,  abc. 

The  number  of  their  combinations  is  16  (2*),  this  is  also 
the  number  of  the  causes  sought,  which  are: 

1.  {abc -{- abc  -\- ab' c -V  ah  c  -\- abc  =  o) 

==  (a  +  be  =  o)  =  (a  =  o)  {b<^c); 

2.  {abc  ■\- ab  c  +  ab  c  -\- abc -\- ab c  >=  0) 

=  {abc  +  ab'  -j-  a  b  =  o)  =  {ab<^c)  {a  =  b); 

5* 


-68  LAW    OF    CAUSES. 

3.  {abc  -\-  ab'  c  -V  a  b  c  -^abc-Vabc^o) 

=  {be  +  b'c  +  ab' c  ==  o)  =  {b  =  c)  {a<Cb  -\-  c); 

4.  {abc  +  ab' c  +  a b' c  +  abc  +  a  b' c  ==  o) 

=  (/  -\- a^  =  o)  =  {c  =  1)  {a<^ b); 

5.  {abc -^  abc  -\- abc  ■\- abc  ■\- abc  +  abc  =0) 

=  {a  +  b  =  o)  =  {a  =  o)  {b  '=  o); 

6.  {abc -\- abc  +  ab  c  +  ab  c  -\- a  b c  -\-  a  b  c  =  o) 

^  {a  ■\-  be  •\-  b' c  =  o)  =  {a  =  o)  {b  =  c); 

7.  {abc  +  abc  +  abc  -\-  ab  c  -\-  a  be  +  a  b  c'  ==  o) 

=  (d!+  /=  o)  =  (ar  ==  o)   (<:=  i)^; 

8.  {abc  +  ab  e  +  ab  c  +abc+abc'+abc  =  o) 

=  {ac  +ac+abc  +  abc  =  o) 

=  {a  ==  c)  {ac <ib<^a  -\-  c)  ^  {a  =  b  ==  c); 

9.  {abc  +  abc^abc  -\-abc-\-abc-\-abc  =0) 

=  {c  -\-  ab  +  a  b  =  o)  =  {c  =  1)  {a  =  b); 

10.  {abc  -^  ab' c  -{■  ab'c  +  a' b c  ■\-  a' b' c  +  a  b' c  =  o) 

=  (^'  +  /  =  o)  =  (^  «=  <r  =  i). 

Before  going  any  further,  it  may  be  observed  that  when 
the  sum  of  certain  constituents  is  equal  to  o,  the  sum  of 
the  rest  is  equal  to  i.  Consequently,  instead  of  examining 
the  sum  of  seven  constituents  obtained  by  ignoring  one  of 
the  four  missing  constituents,  we  can  examine  the  equalities 
obtained  by  equating  each  of  these  constituents  to  i : 

1 1.  {a  b'c  =  x)  =  {a  -\-  b  -^^  c  =  0)  =  {a  =  b  =  c  =  o); 

12.  {a' b' c  =  1)  =  {a  -{■  b  -\-  c  =  0)  =  {a  =  b  =  o)  {c  =  i); 

13.  {a'bc  =  1)  =  {a  +  b'  +  c  —  o)  =  {a  =  o)  {b  =  c  =  i); 

14.  {abc  =1)  ^^  {a  =  b  ==  c  =  i). 


I  It  will  be  observed  that  this  cause  is  the  only  one  which  is  inde- 
pendent of  d;  and  indeed,  in  this  case,  whatever  d  is,  it  will  always 
contain  a  and  will  always  be  contained  in  c.  Compare  Cause  5,  which 
is  independent  of  c,  and  Cause  10,  which  is  independent  of  a. 


FORMS  OF  CONSEQUENCES  AND  CAUSES.  69 

Note  that  the  last  four  causes  are  based  on  the  inclusion 
o<i. 

The  last  two  causes  (15.  and  16.)  are  obtained  either  by 
adding  all  the  missing  constituents  or  by  not  adding  any. 
In  the  first  case,  the  sum  of  all  the  constituents  being  equal 
to  I,  we  find 

15.  1=0, 

that  is,  absurdity,  and  this  confirms  the  paradoxical  prop- 
osition that  the  false  (the  absurd)  implies  any  proposition 
(is  its  cause).  In  the  second  case,  we  obtain  simply  the 
given  equality,  which  thus  appears  as  one  of  its  own  causes 
(by  the  principle  of  identity): 

16.  ab'  -\-  be  =  o. 

If  we  disregard  these  two  extreme  causes,  the  number  of 
causes  properly  so  called  will  be 

22''-'«  —  2. 

46.   Forms    of   Consequences   and    Causes.— We   can 

apply  the  law  of  forms  to  the  consequences  and  causes  of  a 
given  equality  so  as  to  obtain  all  the  forms  possible  to  each 
of  them.  Since  any  equality  is  equivalent  to  one  of  the  two  forms 

each  of  its  consequences  has  the  form* 

NX^-o,         ox  N"  ^  X' ==  \, 

and  each  of  its  causes  has  the  form 

N -^r  X=o,         or  N'X'  =  I. 

*  In  S  44  we  said  that  a  consequence  is  obtained  by  taking  a  part 
of  the  constituents  of  the  first  member  N,  and  not  by  multiplying  it  by 
a  term  X;  but  it  is  easily  seen  that  this  amounts  to  the  same  thing. 
For,  suppose  that  X  (like  N)  be  developed  with  respect  to  the  n  terms 
of  discourse.  It  will  be  composed  of  a  certain  number  of  constituents. 
To  perform  the  multiplication  of  N  by  X,  it  is  sufficient  to  multiply 
all  their  constituents  each  by  each.  Now,  the  product  of  two  identical 
constituents  is  equal  to  each  of  them,  and  the  product  of  two  diflferent 
constituents  is  o.  Hence  the  product  of  yV  by  JiT  becomes  reduced  to 
the  sum  of  the  constituents  common  to  N  and  X,  which  is,  of  course, 
contained  in  N.  So,  to  multiply  N  by  an  arbitrary  term  is  tantamount 
to  taking  a  part  of  its  constituents  (or  all,  or  none). 


yO  VENN  S   PROBLEM.  -^d^ 

y 
In  fact,  we  have  the  following  formal  implications; 

(iV^  +  ^  =  o)<  (iV^  =  o)  <{NX  =  o), 

{N'X'  =   i)<  (i/  =  i)  =  (7/  +  J^'  =  i). 

Applying  the  law  of  forms,  the  formula  of  the  conse- 
quences becomes 

U=  (i\^+  X')  U\  NX  I/, 

and  the  formula  of  the  causes 

U='  N'x'  U  -\-  {N-\-  X)  I/; 

or,  more  generally,  since  X  and  X  are  indeterminate  terms, 
and  consequently  are  not  necessarily  the  negatives  of  each 
other,  the  formula  of  the  consequences  will  be 

U=  {N'  ■\-  X)  U  -\-  NYU', 

and  the  formula  of  the  causes 

U=  N'XU-V  {N\  Y)U' . 

The  first  denotes  that  U  is  contained  in  {N"  +  X)  and 
contains  N  Y\  which  indeed  results,  a  fortiori,  from  the  hypoth- 
esis that    U  is  contained  in  N"  and  contains  N. 

The  second  formula  denotes  that  U  is  contained  in  N' X 
and  contains  N'  +  Y  whence  results,  a  fortiori,  that  U  is 
contained  in  N'  and  contains  N. 

We  can  express  this  rule  verbally  if  we  agree  to  call 
every  class  contained  in  another  a  sub -class,  and  every 
class  that  contains  another  a  super-class.  We  then  say: 
To  obtain  all  the  consequences  of  an  equality  (put  in  the 
form  U  =  N"  U  ■\-  N  U' ),  it  is  sufficient  to  substitute  for  its 
logical  whole  N"  all  its  super-classes,  and,  for  its  logical 
zero  N,  all  its  sub-classes.  Conversely,  to  obtain  all  the 
causes  of  the  same  equality,  it  is  sufficient  to  substitute  for 
its  logical  whole  all  its  sub -classes,  and  for  its  logical  zero, 
all  its  super-classes. 

-47.  Example:  Venn's  Problem. — The  members  of  the 
administrative  council  of  a  financial  society  are  either  bond- 
holders   or   shareholders,    but   not   both.      Now,    all  the  bond- 


-f 


Venn's  problem.  71 

holders  form    a  part   of  the   council.      What  conclusion  must 
we  draw  ? 

Let  a  be  the  class  of  the  members  of  the  council;  let  b 
be  the  class  of  the  bondholders  and  c  that  of  the  share- 
holders. The  data  of  the  problem  may  be  expressed  as 
follows : 

a<^bc-^^c,         b<ia. 

Reducing  to  a  single  developed  equality, 

\. a  {be  =  b  c)  ^  o,         a  b  ^  o, 

(i)  abc  ■\- ab  c  -\- abc  +  a  be  =0. 

This  equality,  which  contains  4  of  the  constituents,  is 
equivalent  to  the  following,  which  contains  the  four  others, 

(2)  abc  -\-ab'e-\-ab'c-\-abe  =  1. 

This  equality  may  be  expressed  in  as  many  different  forms 
as  there  are  classes  in  the  universe  of  the  three  terms 
a,  b,  c. 

Ex.   I.  a  =  abc-\-abe-\-abc-\-abc, 

that  is, 

b  <ia<^bc  -\-  b'c, 

Ex.  2.  b  =  abc  +  ab  c  =  ae  ; 

Ex.  3.  c  =  ab'  e  -\-  a  b'  e  +  ab'  c'  ■\-  abc 


that  is, 


ab' -\-  a' b<Cc<i^ . 


These  are  the  solutions  obtained  by  solving  equation  (i) 
with  respect  to  a,  b,  and  c. 

From  equality  (i)  we  can  derive  16  consequences  as 
follows : 

1.  abc  =0: 

2.  {ab  c  =  o)  =  {a<ib  -\-  e); 

3.  {a' be  =  o)  =^  {bc<ia); 

4.  {abc  =  o)  =  {b  <i  a  -{■  c); 


72  Venn's  problem. 

5.  {abc  +  al)  c  ==  o)  =  {a  <^bc  -{-be)  [ist  premise]; 

6.  {abc  -\-  a  be  =  o)  =  {be  =  o); 

7.  {abc  -\-  a  be'  =  o)  =  {b  <^  ac  -\-  a  e); 

8.  {ab'c -\- abe=  0)  =  {bc<Ca<ib  ■\- c); 

9.  (a^/  -\-a'bc'  ==  o)  =  (a^'  +i2'(^  <;^  ^); 

10.  {a  be -\-  dbe  ==  o)  =  {a  b  =  o)  {2^  premise]; 

11.  {abe-\- ab  e  -\- abc  =  6)  =  {be-\-abc  =  o); 

12.  abe-\- ab  e  -{■  abe  =  o; 

13.  (a^^  ■\-  a  be  -\-  a  be  =  o)  =  {be  +  a  be  )  =  o; 

14.  ab  c  +  a  be  +  a  be  =  o. 

The  last  two  consequences,  as  we  know,  are  the  identity 
(o  =  o)  and  the  equality  (i)  itself.  Among  the  preceding 
consequences  will  be  especially  noted  the  6'^  (^^=0),  the 
resultant  of  the  elimination  of  a,  and  the  loth  {a'b  =  o), 
the  resultant  of  the  elimination  of  e.  When  b  is  eliminated 
the  resultant  is  the  identity 

[{a  +  c)  ae  =  o]  =  {o  =  o). 

Finally,  we  can  deduce  from  the  equality  (i)  or  its  equiv- 
alent (2)  the  following  16  causes: 

1.  {abc  =  i)  =  {a  =  1)  {b=  i)  {c  =  o); 

2.  {ab'c  =  i)  =  {a  =  1)  {b  =  o)  {e  =  i); 

3.  {ab'c  =  1)  =  (a  =  o)  (<J  =  o)  {c  =  i); 
4-  {a  b  e  =  i)  <=  {a  =  o)  {b  =0)  (^  ==  o); 

5.  {abc  +  ab' c  =  1)  =  {a  =  1)  {b'  =  c); 

6.  {abe  +  a  b'e  =  1)  =  {a  =^  b  =  c); 

7.  {abe  +  a'b'c=  i)  =  {c  =  o)  {a  =  b); 

8.  {ab'c+  ab'c  =  i)  =  {b  =  o)  (c  =  i); 
9-  {ab'e+  a  i c  =  i)  =-  {b  •=  o)  {a  =  e); 

10.  {a'b'e+  a  b' c  =  i)  =  (^  =  o)  {b  =  o); 


GEOMETRICAL   DIAGRAMS    OF   VENN.  73 

11.  {abc  +  ab'  c  +  a  b'  c  =  i)  =  {b  =  c)  {c  <^  a); 

12.  {abc  +  ab'c  +a  b'c  =  i)  =  (be  =■  o)  (a  =  ^  +  ^); 

13.  (dr^/  +  a  b' c-\-  a  b' c  =  i)  =  {ac  =  0)  {a  =  b); 

14.  {ab  c  +  a  b  c-\-a  b  c  =  1)^(^=0)  {a<C^c). 

The  last  two  causes,  as  we  know,  are  the  equality  (i) 
itself  and  the  absurdity  (i  =  o).  It  is  evident  that  the 
cause  independent  of  a  is  the  Z^  {b  =  6)  {c  =  \),  and  the 
cause  independent  of  c  is  the  loth  {a  =  o)  {b  ^  o).  There 
is  no  cause,  properly  speaking,  independent  of  b.  The  most 
"natural"  cause,  the  one  which  may  be  at  once  divined 
simply  by  the  exercise  of  common  sense,  is  the   X2^^\ 

{be  =  o)  (a  =  b  +  c). 

But  other  causes  are  just  as  possible;  for  instance  the  9* 
{b  =0)  {a  ==  c),  the  7^11  {c  =0)  (a;  =  b),  or  the  13th 
{ac  =  0)  {a  =  b). 

We  see  that  this  method  furnishes  the  complete  enumera- 
tion of  all  possible  cases.  In  particular,  it  comprises,  among 
the  forms  of  an  equality,  the  solutions  deducible  therefrom 
with  respect  to  such  and  such  an  "unknown  quantity",  and, 
among  the  consequences  of  an  equality,  the  resultants  of  the 
eUmination  of  such  and  such  a  term. 

48.  The  Geometrical  Diagrams  of  Venn. — Poretsky's 
method  may  be  looked  upon  as  the  perfection  of  the  methods 
of  Stanley  Jevons  and  Venn. 

Conversely,  it  finds  in  them  a  geometrical  and  mechanical 
illustration,  for  Venn's  method  is  translated  in  geometrical 
diagrams  which  represent  all  the  constituents,  so  that,  in 
order  to  obtain  the  result,  we  need  only  strike  out  (by 
shading)  those  which  are  made  to  vanish  by  the  data  of  the 
problem.  For  instance,  the  universe  of  three  terms  a,  b,  c, 
represented  by  the  unbounded  plane,  is  divided  by  three 
simple  closed  contours  into  eight  regions  which  represent  the 
eight  constituents  (Fig.  i). 


74 


GEOMETRICAL   DIAGRAMS    OF    VENN. 


ob'b'c' 


Fig.  X. 


To  represent  geometrically  the  data  of  Venn's  problem  we 
must  strike  out  the  regions  abc,  ab  c ,  a  be  and  a  be;  there 
will   then   remain  the   regions   abc ,    abc,    a  b  e,    and    a  b  e 
which    will    constitute    the    universe    relative    to    the  problem, 
being  what  Poretsky  calls  his   logical  whole  (Fig.   2).     Then 


cb'b'c ' 


Fig.  2. 


every  class  will  be  contained  in  this  universe,  which  will  give 
for  each  class  the  expression  resulting  from  the  data  of  the 
problem.  Thus,  simply  by  inspecting  the  diagram,  we  see 
that  the  region  be  does  not  exist  (being  struck  out);  that  the 
region  b  is  reduced  to  abc  (hence  to  ab)\  that  all  a  v^  b 
or  c,  and  so  on. 


LOGICAL   MACHINE   OF   JEVONS.  75 

This  diagrammatic  method  has,  however,  serious  incon- 
veniences as  a  method  for  solving  logical  problems.  It  does 
not  show  how  the  data  are  exhibited  by  canceling  certain 
constituents,  nor  does  it  show  how  to  combine  the  remaining 
constituents  so  as  to  obtain  the  consequences  sought.  In 
short,  it  serves  only  to  exhibit  one  single  step  in  the  argument, 
namely  the  equation  of  the  problem;  it  dispenses  neither  with 
the  previous  steps,  /*.  e.,  "throwing  of  the  problem  into  an 
equation"  and  the  transformation  of  the  premises,  nor  with 
the  subsequent  steps,  /.  e.,  the  combinations  that  lead  to 
the  various  consequences.  Hence  it  is  of  very  little  use, 
inasmuch  as  the  constituents  can  be  represented  by  algebraic 
symbols  quite  as  well  as  by  plane  regions,  and  are  much 
easier  to  deal  with  in  this  form. 

49.  The  Logical  Machine  of  Jevons. — In  order  to 
make  his  diagrams  more  tractable,  Venn  proposed  a  me- 
chanical device  by  which  the  plane  regions  to  be  struck  out 
could  be  lowered  and  caused  to  disappear.  But  Jevons 
invented  a  more  complete  mechanism,  a  sort  of  logical  piano. 
The  keyboard  of  this  instrument  was  composed  of  keys  in- 
dicating the  various  simple  terms  {a,  b,  c,  d),  their  negatives, 
and  the  signs  -f  and  =.  Another  part  of  the  instrument 
consisted  of  a  panel  with  movable  tablets  on  which  were 
written  all  the  combinations  of  simple  terms  and  their  neg- 
atives; that  is,  all  the  constituents  of  the  universe  of  dis- 
course. Instead  of  writing  out  the  equalities  which  represent 
the  premises,  they  are  "played"  on  a  keyboard  like  that  of 
a  typewriter.  The  result  is  that  the  constituents  which  vanish 
because  of  the  premises  disappear  from  the  panel.  When 
all  the  premises  have  been  "played",  the  panel  shows  only 
those  constituents  whose  sum  is  equal  to  r,  that  is,  forms 
the  universe  with  respect  to  the  problem,  its  logical  whole. 
This  mechanical  method  has  the  advantage  over  Venn's  geo- 
metrical method  of  performing  automatically  the  "throwing 
into  an  equation",  although  the  premises  must  first  be  ex- 
pressed in  the  form  of  equalities;  but  it  throws  no  more  hght 
than    the    geometrical    method  on  the  operations  to  be  per- 


y6  TABLE    OF    CONSEQUENCES. 

formed    in    order   to    draw    the   consequences  from  the  data 
displayed  on  the  panel. 

50.  Table  of  Consequences. — But  Poretsky's  method 
can  be  illustrated,  better  than  by  geometrical  and  mechanical 
devices,  by  the  construction  of  a  table  which  will  exhibit 
directly  all  the  consequences  and  all  the  causes  of  a  given 
equality.  (This  table  is  relative  to  this  equality  and  each 
equality  requires  a  different  table).  Each  table  comprises 
the  2«  classes  that  can  be  defined  and  distinguished  in  the 
universe  of  discourse  of  n  terms.  We  know  that  an  equality 
consists  in  the  annulment  of  a  certain  number  of  these 
classes,  viz.,  of  those  which  have  for  constituents  some  of 
the  constituents  of  its  logical  zero  N.  Let  m  be  the  number 
of  these  latter  constituents,  then  the  number  of  the  sub- 
classes of  iV  is  2 »'  which,  therefore,  is  the  number  of  classes 
of  the  universe  which  vanish  in  consequence  of  the  equality 
considered.  Arrange  them  in  a  column  commencing  with 
o  and  ending  with  N  (the  two  extremes).  On  the  other 
hand,  given  any  class  at  all,  any  preceding  class  may  be 
added  to  it  without  altering  its  value,  since  by  hypothesis 
they  are  null  (in  the  problem  under  consideration).  Conse- 
quently, by  the  data  of  the  problem,  each  class  is  equal  to 
2W  classes  (including  itself).  Thus,  the  assemblage  of  the 
2«  classes  of  discourse  is  divided  into  2"-'''  series  of  2^ 
classes,  each  series  being  constituted  by  the  sums  of  a  certain 
class  and  of  the  2*"^  classes  of  the  first  column  (sub-classes 
of  N).  Hence  we  can  arrange  these  2 '«  sums  in  the 
following  columns  by  making  them  correspond  horizontally 
to  the  classes  of  the  first  column  which  gave  rise  to  them. 
Let  us  take,  for  instance,  the  very  simple  equality  a  =  b, 
which  is  equivalent  to 

ab   +  a  b  =^  o. 

The  logical  zero  (JV)  in  this  case  is  ab'  -\-  ab.  It  com- 
prises two  constituents  and  consequently  four  sub-classes: 
o,  ab,  a  b,  and  ab'  -\-  a  b.  These  will  compose  the  first 
column.    The  other  classes  of  discourse  are  ab^  a  b' ,  ab  +  a  b  , 


TABLE    OF    CAUSES.  "Jl 

and    those    obtained  by   adding    to  each    of  them  the   four 

classes  of  the  first  column.     In  this  way,  the  following  table 
is  obtained: 

o  ab         a  b  ab  -\-  a  b 

ab  a            b  a  ■\-  b 

a  b  b             a  a  +  b 

ab  +  a  b  a  +  b     a  +  b  i 

By  construction,  each  class  of  this  table  is  the  sum  of 
those  at  the  head  of  its  row  and  of  its  column,  and,  by  the 
data  of  the  problem,  it  is  equal  to  each  of  those  in  the 
same  column.  Thus  we  have  64  different  consequences  for 
any  equality  in  the  universe  of  discourse  of  2  letters.  They 
comprise  16  identities  (obtained  by  equating  each  class  to 
itself)  and  i6  forms  of  the  given  equality,  obtained  by 
equating  the  classes  which  correspond  in  each  row  to  the 
classes  which  are  known  to  be  equal  to  them,  namely 

o  =  ab  +  a  b,     ab  =  a  +  b,     a  b  =  a  +  b  ,     ab  +  a  b  =  1 

a  =  b,  b'  ==  a',         ab'  =  a  b^         a-\-  b  =  a  ■\-  b. 

Each  of  these  8  equalities  counts  for  two,  according  as  it 
is  considered  as  a  determination  of  one  or  the  other  of  its 
members. 

51.  Table  of  Causes. — The  same  table  may  serve  to 
represent  all  the  causes  of  the  same  equality  in  accordance 
with  the  following  theorem: 

When  the  consequences  of  an  equality  iV  =  o  are  ex- 
pressed in  the  form  of  determinations  of  any  class  U,  the 
causes  of  this  equality  are  deduced  from  the  consequences 
of  the  opposite  equality,  JV"=  i,  put  in  the  same  form, 
by  changing    U  to  if  in  one  of  the  two  members. 

For  we  know  that  the  consequences  of  the  equality  N=  o 
have  the  form 

U^iN"  -^  X)  U  ^  NYlf, 
and  that  the  causes   of  the  same  equality  have  the  form 
U  =  l/X  U  ^  {N^  Y)  [/. 


78  TABLE    OF    CAUSES. 

Now,  if  we  change    U  into    U    in  one  of  the  members  o 
this  last  formula,  it  becomes 

U^{N  ^  X!)  U  ^  N'Y  U\ 

and  the  accents  of  X  and  Y  can  be  suppressed  since  these 
letters  represent  indeterminate  classes.  But  then  we  have 
the  formula  of  the  consequences  of  the  equality  N'  =  o  or 
N^  I. 

This  theorem  being  established,  let  us  construct,  for  in- 
stance, the  table  of  causes  of  the  equality  a  =  b.  This  will 
be  the  table  of  the  consequences  of  the  opposite  equality 
a  =  i ^  for  the  first  is  equivalent  to 

ab  -\-  a  b  =  o^ 

and  the  second  to 

{ab  ■\-  a  b'  =  o)  =  {ab'  ■\-  a'b  ^^  i). 

o                ab  a  b  ab  +  a  b 

ab                 a  b  a  -\-  b 

a  b                b  a  a   -^  b 

ab  ■\-  al?        a  -\-  b'     a  +  b  i 

To  derive  the  causes  of  the  equality  a  =  b  from  this  table 
instead  of  the  consequences  of  the  opposite  equality  a  =  b' , 
it  is  sufficient  to  equate  the  negative  of  each  class  to  each 
of  the  classes  in  the  same  column.     Examples  are: 

a'  ■\-  ^  =  o,  a  -\-  b  =  a  b  ,  a  ■\-  b'  =  ab  ■\-  a  b' , 

a  -\-  b  =  a,  a   ■\-  b  =  b' ,  a  -\-  b  =  a  ■\-  b' ; . . . . 

Among  the  64  causes  of  the  equality  under  consideration 
there  are  16  absurdities  (consisting  in  equating  each  class  of 
the  table  to  its  negative);  and  16  forms  of  the  equality  (the 
same,  of  course,  as  in  the  table  of  consequences,  for  two 
equivalent  equalities  are  at  the  same  time  both  cause  and 
consequence  of  each  other). 

It  will  be  noted  that  the  table  of  causes  differs  from  the 
table  of  consequences  only  in  the  fact  that  it  is  sym- 
metrical to  the  other  table  with  respect  to  the  principal  diagonal 


NUMBER    OF   POSSIBLE   ASSERTIONS,  79 

(o,  i);  hence  they  can  be  made  identical  by  substituting  the 
word  "row"  for  the  word  "column"  in  the  foregoing  state- 
ment. And,  indeed,  since  the  rule  of  the  consequences  con- 
cerns only  classes  of  the  same  column,  we  are  at  liberty  so  to 
arrange  the  classes  in  each  column  on  the  rows  that  the 
rule  of  the  causes  will  be  verified  by  the  classes  in  the 
same  row. 

It  will  be  noted,  moreover,  that,  by  the  method  of  con- 
struction adopted  for  this  table,  the  classes  which  are  the 
negatives  of  each  other  occupy  positions  symmetrical  with 
respect  to  the  center  of  the  table.  For  this  result,  the  sub- 
classes of  the  class  JV'  (the  logical  whole  of  the  given 
equality  or  the  logical  zero  of  the  opposite  equality)  must 
be  placed  in  the  first  row  in  their  natural  order  from  o  to  J\  ; 
then,  in  each  division,  must  be  placed  the  sum  of  the  classes 
at  the  head  of  its  row  and  column. 

With  this  precaution,  we  may  sum  up  the  two  rules  in  the 
following  practical  statement: 

To  obtain  every  consequence  of  the  given  equality  (to 
which  the  table  relates)  it  is  sufficient  to  equate  each  class 
to  every  class  in  the  same  column;  and,  to  obtain  every 
cause,  it  is  sufficient  to  equate  each  class  to  every  class  in 
the  row  occupied  by  its  symmetrical  class. 

It  is  clear  that  the  table  relating  to  the  equality  iV  =  o 
can  also  serve  for  the  opposite  equality  JV  ^  i,  on  condition 
that  the  words  "row"  and  "column"  in  the  foregoing  statement 
be  interchanged. 

Of  course  the  construction  of  the  table  relating  to  a  given 
equality  is  useful  and  profitable  only  when  we  wish  to 
enumerate  all  the  consequences  .or  the  causes  of  this  equal- 
ity. If  we  desire  only  one  particular  consequence  or  cause 
relating  to  this  or  that  class  of  the  discourse,  we  make  use 
of  one  of  the  formulas  given  above. 

52.  The  Number  of  Possible  Assertions. — If  we  regard 
logical  functions  and  equations  as  developed  with  respect  to 
all  the  letters,  we  can  calculate  the  number  of  assertions  or 
different   problems   that    may  be   formulated    about  n  simple 


8o  PARTICULAR    PROPOSITIONS. 

terms.  For  all  the  functions  thus  developed  can  contain  only 
those  constituents  which  have  the  coefficient  i  or  the  coef- 
ficient o  (and  in  the  latter  case,  they  do  not  contain  them). 
Hence  they  are  additive  combinations  of  these  constituents; 
and,  since  the  number  of  the  constituents  is  2^,  the  number 
of  possible  functions  is  2*".  From  this  must  be  deducted 
the  function  in  which  all  constituents  are  absent,  which  is 
identically  o,  leaving  2^"— i  possible  equations  (255  when 
«  =  3).  But  these  equations,  in  their  turn,  may  be  combined 
by  logical  addition,  /.  <?.,  by  alternation;  hence  the  number 
of  their  combinations  is  22^"— i_i^  excepting  always  the 
null  combination.  This  is  the  number  of  possible  assertions 
affecting  n  terms.  When  «  =  2,  this  number  is  as  high  as 
32767.^  We  must  observe  that  only  universal  premises  are 
admitted  in  this  calculus,  as  will  be  explained  in  the  follow- 
ing section. 

53.  Particular  Propositions. — Hitherto  we  have  only 
considered  propositions  with  an  affirmative  copula  {i.  e.,  in- 
clusions or  equalities)  corresponding  to  the  universal  prop- 
ositions of  classical  logic.^  It  remains  for  us  to  study  prop- 
ositions with  a  negative  copula  (non  inclusions  or  inequalities), 
which  translate  particular  propositions  3;    but  the  calculus  of 

1  G.  Peano,  Calcob  geometrico  (1888)  p.  x;  Schr6der,  Algebra  der 
Logik,  Vol.  II,  p.  144—148. 

2  The  universal  affirmative,  "All  a's  are  ^'s",  may  be  expressed  by 
the  formulas 

(a  <Cb)^={a  =  ab)  =  (ab'=  o)  =  (o'  -j-  3  =  l), 

and  the  universal  negative,  "No  a's  are  ^'s",  by  the  formulas 

(a  <  <^')=  {a  =  ab')  =  {ab  =  o)  =  (a  -f  <5'=  l). 

3  For  the  particular  affirmative,  "Some  a's  are  (5's",  being  the  negation 
of  the  universal  negative,  is  expressed  by  the  formulas 

(a  <^  b')  =  (a  =f=  ab')  =  (ab  4=  o)  =  («'+  b' ^  \), 

and  the  particular  negative,  "Some  a's  are  not  b''%",  being  the  negation 
of  the  universal  affirmative,  is  expressed  by  the  formulas 

{a<i^b)^[a^  ab)  =  {ab' ^  o)  =  (a'+  3  =}=  l). 


SOLUTION    OF    AN    INEQUATION    WITH    ONE   UNKNOWN.         8 1 

propositions  having  a  negative  copula  results  from  laws  al- 
ready known,  especially  from  the  formulas  of  De  Morgan 
and  the  law  of  contraposition.  We  shall  enumerate  the  chief 
formulas  derived  from  it. 

The  principle  of  composition  gives  rise  to  the  following 
formulas: 

{c<ab)  =  {c<^a)  +  (^<^), 

whence  come  the  particular  instances 

(a  +  ^  4=  o)  =  (a  4=  o)  +  (^  +  o). 

From  these  may  be  deduced  the  following  important  im- 
plications: 

(«  +  o)<  (a  +  /5  +  o), 

From  the  principle  of  the  syllogism,  we  deduce,  by  the 
law  of  transposition, 

{a<b)  (a+o)<(/J  +  o), 

{a<b)  (^+i)<(a+i). 

The  formulas  for  transforming  inclusions  and  equalities 
give  corresponding  formulas  for  the  transformation  of  non- 
inclusions  and  inequalities, 

{a<^b)=^  {ai^  o)  =  (a'  +  ^  4=  i), 

(a  4=  ^)  =  {ai  -V  db  4=  o)     =  {ab  +  a^'+  i). 

54.  Solution  of  an  Inequation  with  One  Unknown. — 

If  we  consider  the  conditional  inequality  {inequation)  with 
one  unknown 

ax  ^  bx  -^  o^ 

we  know  that  its  first  member  is  contained  in  the  sum  of 
its  coefficients 

ax  ^r  bx  <^a  ^r  b. 


82  SOLUTION    OF    AN   INEQUATION    WITH    ONE   UNKNOWN. 

From  this  we  conclude  that,  if  this  inequation  is  verified, 
we  have  the  inequality 

a  +  d  ^  o. 

This  is  the  necessary  condition  of  the  solvability  of  the 
inequation,  and  the  resultant  of  the  elimination  of  the  un- 
known X.     For,  since  we  have  the  equivalence 

Y\  {ax  +  <^^  «=  o)  =  (a  +  /^  =  o), 

X 

we  have  also  by  contraposition  the  equivalence 
"V  {ax  -V  b X  =^  o)  =  {a  ■\-  b  ^  6). 

Likewise,  from  the  equivalence 

2  {ax  +  bx  =  o)  ^=  {ab  =  o), 

X 

we  can  deduce  the  equivalence 

J~][  {ax  +  bx  =4=  o)  =  {ab  =4=  o), 

X 

which  signifies  that  the  necessary  and  sufficient  condition  for 
the  inequation  to  be  always  true  is 

(a/^  +  o); 
and,  indeed,  we  know  that  in  this  case  the  equation 

{ax  A:  bx  =  d) 
is  impossible  (never  true). 

Since,  moreover,  we  have  the  equivalence 

{ax  +  bx  =  o)  =  {x  =  a  X  +  bx  ), 
we  have  also  the  equivalence 

{ax  +  bx'  =^  o)  =  {x=^  a  X  +  bx'). 
Notice  the  significance  of  this  solution: 
{ax  +  bx'^  o)  =  {ax  +  o)  +  {bx  4=  o)  =  (a:  <}:  a)  +  {b<i^x). 

"Either  x  is  not  contained  in  a  ,  or  it  does  not  contain  b". 
This  is  the  negative  of  the  double  inclusion 

b<^x<Ca  ' 


EQUATION    AND    AN    INEQUATION.  83 

Just  as  the  product  of  several  equalities  is  reduced  to  one 
single  equality,  the  sum  (the  alternative)  of  several  inequalities 
may  be  reduced  to  a  single  inequality.  But  neither  several 
alternative  equalities  nor  several  simultaneous  inequalities  can 
be  reduced  to  one. 

55.   System  of  an  Equation  and  an  Inequation. — We 

shall    limit  our   study  to  the  case  of  a  simultaneous  equality 
and  inequality.     For  instance,  let  the  two  premises  be 

{ax  +  bx  =  o)  {ex  +  dx  +  o). 

To  satisfy  the  former  (the  equation)  its  resultant  ab  =  o 
must  be  verified.     The  solution  of  this  equation  is 

x  =  a  X  -^^  bx  . 

Substituting  this  expression  (which  is  equivalent  to  the 
equation)  in  the  inequation,  the  latter  becomes 

{a'c  +  ad)x  +  {be  +  b'd)x'  ^  o. 

Its  resultant  (the  condition  of  its  solvability)  is 

{a'e  ^  ad  +  be  +  b'd^o)  =  [(a' +  b)e  -\-  {a  +  b')d^o\, 

which,  taking  into  account  the  resultant  of  the  equality, 

{ab  =  o)  =  {a'  +  b  =  a)  =  {a  +  b'  =  B') 

may  be  reduced  to 

a  e  +  b  d=^  o. 

The  same  result  may  be  reached  by  observing  that  the 
equality  is  equivalent  to  the  two  inclusions 

{x<ia')  {x<ib'), 

and  by  multiplying  both  members  of  each  by  the  same  term 

{ex<.  a'e)  {dx  <  b' d)  <  (r;c  +  </;c'  <  ae  +  b' d) 

{ex  +  dx  +  o)<C  (^'^  +  b'd^  o). 

This  resultant  implies  the  resultant  of  the  inequality  taken 

alone 

e  +  d^o, 

so   that  we  do  not  need  to  take  the  latter  into  account.     It 

6* 


84  CALCULUS    OF    PROPOSITIONS. 

is  therefore  sufficient  to  add  to  it  the  resultant  of  the  equality 
to  have  the  complete  resultant  of  the  proposed  system 

{ab  =  o)   {a  c  +  }>  d-^  o). 

The  solution  of  the  transformed  inequality  (which  conse- 
quently involves  the  solution  of  the  equality)  is 

x^{ac'  -V  ad')x  +  (be  +  b'd^x. 

56.  Formulas  Peculiar  to  the  Calculus  of  Propositions. 

— All  the  formulas  which  we  have  hitherto  noted  are  valid 
alike  for  propositions  and  for  concepts.  We  shall  now 
establish  a  series  of  formulas  which  are  valid  only  for  prop- 
ositions, because  all  of  them  are  derived  from  an  axiom 
peculiar  to  the  calculus  of  propositions,  which  may  be  called 
-the  principle  of  assertion. 

This  axiom  is  as  follows: 

(Ax.  X.)  (a  =  i)  =  a. 

P.  I.:  To  say  that  a  proposition  a  is  true  is  to  state  the 
proposition  itself.  In  other  words,  to  state  a  proposition  is 
to  affirm  the  truth  of  that  proposition.^ 

Corollary: 

«'=  {a  =  i)  =  (a  =  o). 

P.  L:  The  negative  of  a  proposition  a  is  equivalent  to  the 
affirmation  that  this  proposition  is  false. 

By  Ax.  IX  (§  20),  we  already  have 

{a  =  i)  (a  =  o)  =  o, 

"A  proposition  cannot  be  both  true  and  false  at  the  same 
time",  for 

(Syll.)  (a  =  I)  (^  =  o)<  (I  =  o)  =  o. 


I  We  can  see  at  once  that  this  formula  is  not  susceptible  of  a  con- 
ceptual interpretation  (C.  I.);  for,  if  a  is  a  concept,  (a  =  I)  is  a  prop- 
osition, and  we  would  then  have  a  logical  equality  (identity)  between 
a  concept  and  a  proposition,  which  is  absurd. 


IMPLICATION    AND    ALTERNATIVE.  85 

But  now,  according  to  Ax.  X,  we  have 

{a  =  i)  +  {a  =  o)  =  a  +  a  =1. 

"A  proposition  is  either  true  or  false".  From  these  two 
formulas  combined  we  deduce  directly  that  the  propositions 
(a  =  i)  and  (a  =  o)  are  contradictory,  t.  e., 

(a  4=  i)  =  (a  =  o),  (a  =f  o)  =  (fl  =  i). 

From  the  point  of  view  of  calculation  Ax.  X  makes  it 
possible  to  reduce  to  its  first  member  every  equality  whose 
second  member  is  i,  and  to  transform  inequalities  into 
equalities.  Of  course  these  equalities  and  inequalities  must 
have  propositions  as  their  members.  Nevertheless  all  the 
formulas  of  this  section  are  also  valid  for  classes  in  the 
particular  case  where  the  universe  of  discourse  contains  only 
one  element,  for  then  there  are  no  classes  but  o  and  i.  In 
short,  the  special  calculus  of  propositions  is  equivalent  to  the 
calculus  of  classes  when  the  classes  can  possess  only  the 
two  values  o  and  i. 

57.  Equivalence  of  an  Implication  and  an  Alternative. 

— The  fundamental  equivalence 

(a  <  ^)  =  («'  +  ^  =  i) 
gives  rise,  by  Ax.  X,  to  the  equivalence 

which  is  no  less  fundamental  in  the  calculus  of  propositions. 
To  say  that  a  implies  l>  is  the  same  as  affirming  "not-<z  or 
d",  i.  e.,  "either  a  is  false  or  b  is  true."  This  equivalence 
is  often  employed  in  every  day  conversation. 

Corollary. — For  any  equality,  we  have  the  equivalence 

{a  =  b)  ^  ab  -\-  a  b  . 

Demonstration : 
(^a  =  b)=^{a<  b)  (b <a)  =  (a  +  b)  (b'  +  a)^ab  +  ab\ 

"To  affirm  that  two  propositions  are  equal  (equivalent) 
is  the  same  as  stating  that  either  both  are  true  or  both  are 
false". 


80  IMPLICATION    AND    ALTERNATIVE. 

The  fundamental  equivalence  established  above  has  im- 
portant consequences  which  we  shall  enumerate. 

In  the  first  place,  it  makes  it  possible  to  reduce  secondary, 
tertiary,  etc.,  propositions  to  primary  propositions,  or  even 
to  sums  (alternatives)  of  elementary  propositions.  For  it 
makes  it  possible  to  suppress  the  copula  of  any  proposition, 
and  consequently  to  lower  its  order  of  complexity.  An  im- 
plication (A  <C  -B),  in  which  A  and  J5  represent  propositions 
more  or  less  complex,  is  reduced  to  the  sum  A^  +  £,  in 
which  only  copulas  within  A  and  B  appear,  that  is,  prop- 
ositions of  an  inferior  order.  Likewise  an  equality  {A  =  B) 
is  reduced  to  the  sum  {AB  ->r  A' J^)  which  is  of  a  lower 
order. 

We  know  that  the  principle  of  composition  makes  it 
possible  to  combine  several  simultaneous  inclusions  or  equal- 
ities, but  we  cannot  combine  alternative  inclusions  or  equal- 
ities, or  at  least  the  result  is  not  equivalent  to  their  alter- 
native but  is  only  a  consequence  of  it.  In  short,  we  have 
only  the  implications 

{a<c)  +  {b<^cX{ab<c\ 
{c<a)  +  {c<bX{c<a^b), 

which,  in  the  special  cases  where  c  =  o  and  ^  =  i,  become 
(a  =  o)  +  (^  =  o)<  (a/J  =  o), 
(a=  I)  +  (/^=  !)<(«  +  ^=  I). 

In  the  calculus  of  classes,  the  converse  implications  are 
not  valid,  for,  from  the  statement  that  the  class  ab  \s,  null, 
we  cannot  conclude  that  one  of  the  classes  a  ox  b  '\i  null 
(they  can  be  not-null  and  still  not  have  any  element  in 
common);  and  from  the  statement  that  the  sum  {a  +  b)  is 
equal  to  i  we  cannot  conclude  that  either  a  or  ^  is  equal 
to  I  (these  classes  can  together  comprise  all  the  elements  of 
the  universe  without  any  of  them  alone  comprising  all).  But 
these  converse  implications  are  true  in  the  calculus  of  prop- 
ositions 

{ab<ic)<{a<c)  -f-  {b<c), 
(^<«  \bX{c<a)^  {c<b)- 


IMPLICATION    AND    ALTERNATIVE.  87 

for  they  are  deduced  from  the  equivalence  established  above, 
or  rather  we  may  deduce  from  it  the  corresponding  equal- 
ities which  imply  them, 

(i)  {ab<c)  =  ia<^c)  +  {b<c), 

(2)  {c<a^  b)  =  {c<:a)  +  {c<b). 

Demonstration: 

(i)  {ab<^c)=^a  -^  b'  ^  c, 

{a<^c)  +  {b<^c)==  {a  +  <:)  +  {b'  -\-  c)  =  a   +  b'  ^  c; 

(2)  (^<a  +  ^)  =  /  +  a  +  ^, 

(^  <  a)  +  (f  <  b)  =  {c'  ■\-  a)  ^-  (c  ■\-  b)  =  c  -\-  a  -V  b. 

In  the  special  cases  where  r  ==  o  and  c  =  i  respectively, 
we  find 

(3)  {ab  =  o)  =  (a  =  o)  +  (^  =  o), 

(4)  (a  +  <^  =  I)  =  (a  =  I)  +  (^  =  i). 

P.  I.:  (i)  To  say  that  two  propositions  united  imply  a 
third  is  to  say  that  one  of  them  implies  this  third  proposition. 

(2)  To  say  that  a  proposition  implies  the  alternative  of 
two  others  is  to  say  that  it  implies  one  of  them. 

(3)  To  say  that  two  propositions  combined  are  false  is  to 
say  that  one  of  them  is  false. 

(4)  To  say  that  the  alternative  of  two  propositions  is  true 
is  to  say  that  one  of  them  is  true. 

The  paradoxical  character  of  the  first  three  of  these  state- 
ments will  be  noted  in  contrast  to  the  self-evident  character 
of  the  fourth.  These  paradoxes  are  explained,  on  the  one 
hand,  by  the  special  axiom  which  states  that  a  proposition 
is  either  true  or  false;  and,  on  the  other  hand,  by  the  fact 
that  the  false  implies  the  true  and  that  only  the  false  is  not 
implied  by  the  true.  For  instance,  if  both  premises  in  the 
first  statement  are  true,  each  of  them  implies  the  conse- 
quence, and  if  one  of  them  is  false,  it  implies  the  conse- 
quence (true  or  false).  In  the  second,  if  the  alternative  is 
true,  one  of  its  terms  must  be  true,  and  consequently  will, 
like  the  alternative,  be  implied  by  the  premise  (true  or  false). 


'88  LAW    OF    IMPORTATION    AND    EXPORTATION. 

Finally,  in  the  third,  the  product  of  two  propositions  cannot 
be  false  unless  one  of  them  is  false,  for,  if  both  were  true, 
their  product  would  be  true  (equal  to   i). 

58.  Law  of  Importation  and  Exportation. — The  funda- 
mental equivalence  (a  <i  d)  =  a  +  b  has  many  other  inter- 
esting consequences.  One  of  the  most  important  of  these 
is  the  law  of  importation  and  exportation,  which  is  expressed 
by  the  following  formula: 

\a<{b<c)\^{ab<c) 

"To  say  that  if  a  is  true  b  implies  c,  is  to  say  that  a 
and  b  imply  ^". 

This  equality  involves  two  converse  implications:  If  we 
infer  the  second  member  from  the  first,  we  import  into  the 
implication  {b<^c)  the  hypothesis  or  condition  a;  if  we  infer 
the  first  member  from  the  second,  we,  on  the  contrary, 
export  from  the  implication  {ab<^e)  the  hypothesis  a. 

Demonstration  : 

[a  <  (iJ  <  <:)]  =  a'  +  (^  <  r)  =  a'  -f  y  +  r, 
(a  /^  <C  ^)  =  (a  3)'  -h  c  =  a  ^r  b'  ■\-  c. 

Cor.   I. — Obviously  we  have  the  equivalence 
\a<{b<c)\  =  [^<(«<^)], 

since  both  members  are  equal  to  {ab<^c),  by  the  commu- 
tative law  of  multiplication. 

Cor.   2. — We  have  also 

[a<{a<b)]  =  {a<b), 
for,  by  the  law  of  importation  and  exportation, 
[a<{a<b)]  =  {aa<b)  =  {a<b). 

If  we  apply  the  law  of  importation  to  the  two  following 
formulas,  of  which  the  first  results  from  the  principle  of 
identity  and  the  second  expresses  the  principle  of  contra- 
position, 


LAW  OF  IMPORTATION  AND  EXPORTATION.        89 

{a<b)<{a<b\         {a<b)<{b'  <a), 

we  obtain  the  two  formulas 

(a<^)a<^,  {a<b)b'<ia', 

which  are  the  two  types  of  hypothetical  reasoning:  "If  a 
implies  b,  and  if  a  is  true,  b  is  true"  {modus  ponens);  "If  a 
implies  b^  and  if  b  is  false,  a. is  false"    {modus  tollens). 

Remark.  These  two  formulas  could  be  directly  deduced 
by  the  principle  of  assertion,  from  the  following 

{a<,b)  (a=  x)<(^=i), 
{a<b)  (^  =  o)<(a  =  o), 

which  are  not  dependent  on  the  law  of  importation  and 
which  result  from  the  principle  of  the  syllogism. 

From  the  same  fundamental  equivalence,  we  can  deduce 
several  paradoxical  formulas: 

1.  a<:{b<.a\         d<:{a<b). 

"If  a  is  true,  a  is  implied  by  any  proposition  b\  if  a  is 
false,  a  implies  any  proposition  ^".  This  agrees  with  the 
known  properties  of  o  and  i. 

2.  «  <  {{a  <b)<  b],         a'<  [{b  <a)<  b'\ 

"If  a  is  true,  then  '<z  imphes  h'  implies  b;  if  a  is  false, 
then  ''b  implies  a'  implies  not-^." 

These  two  formulas  are  other  forms  of  hypothetical  reason- 
ing {modus  ponens  and  modus  tollens). 

3.  {{a  <:bXa-\  =  a\         {{b  <d)<  a]  =  a, 

"To  say  that,  if  a  implies  b,  a  is  true,  is  the  same  as 
affirming  a;  to  say  that,  if  b  implies  a.,  a  is  false,  is  the 
same  as  denying  «". 

Demonstration : 

\(ya  <C^b)  <i_  d\  ==  {a  ■\-  b<^cC)  =  ab  +  a  =  a, 
[(^  <^  a)  <^  dt  ]  =  (y  +  d!  <^  a)  =  a  b  -{■  a  =  a' . 

I  This  formula  is  Bertrand  Russell's  "principle  of  reduction".  See 
The  Principles  of  Mathematics,  Vol.  I,  p.  1 7  (Cambridge,  1 903). 


go  INEQUALITIES   REDUCED   TO    EQUAUTIES. 

In  formulas  (i)  and  (3),  in  which  b  is  any  term  at  all, 
we  might  introduce  the  sign  ]^  with  respect  to  b.  In  the 
following  formula,  it  becomes  necessary  to  make  use  of  this 
sign. 

4.  n  {[«<(''^<^)]<^|  -ab. 

X 

Demonstration : 
{[a<(/^<;c)]<A:}  =  |[a'+  (^<^)]<^| 

==[(a  -{■  b  ■\-  x)<^oc\  =  abx  +  x  =  ab  +  x. 
We    must   now   form    the   product  Y\  (^^  +  ^)i    where  x 

X 

can  assume  every  value,  including  o  and  i.  Now,  it  is 
clear  that  the  part  common  to  all  the  terms  of  the  form 
{ab  +  oc)  can  only  be  ab.  For,  (i)  ab  is  contained  in  each 
of  the  sums  {ab  +  x)  and  therefore  in  the  part  common  to 
all;  (2)  the  part  common  to  all  the  sums  {ab  ^r  x)  must  be 
contained  in  {ab  -\r  o),  that  is,  in  ab.  Hence  this  common 
part  is  equal  to  ab'^,  which  proves  the  theorem. 

59.  Reduction  of  Inequalities  to  Equalities. — As  we 
have  said,  the  principle  of  assertion  enables  us  to  reduce 
inequalities  to  equalities  by  means  of  the  following  formulas: 

(a  +  o)  =  (a  =  i),         (<i  +  i)  =  (a  =  o), 

{a^b)={a==  b'). 
For, 

(a  4=  ^)  =  (a/  +  a'b  +  o)  =  {ab'  +  ah=  i)  =  (a  =  b'). 

Consequently,  we  have  the  paradoxical  formula 

{a^b)  =  {a  =  b'). 

I  This  argument  is  general  and  from  it  we  can  deduce  the  formula 

X 

whence  may  be  derived  the  correlative  formula 


INEQUALITIES    REDUCED    TO    EQUALITIES.  9 1 

This  is  easily  understood,  for,  whatever  the  proposition  b^ 
either  it  is  true  and  its  negative  is  false,  or  it  is  false  and 
its  negative  is  true.  Now,  whatever  the  proposition  a  may 
be,  it  is  true  or  false;  hence  it  is  necessarily  equal  either  to 
b  or  to  iJ .  Thus  to  deny  an  equality  (between  propositions) 
is  to  affirm  the  opposite  equality. 

Thence  it  results  that,  in  the  calculus  of  propositions,  we 
do  not  need  to  take  inequalities  into  consideration — a  fact 
which  greatly  simplifies  both  theory  and  practice.  More- 
over, just  as  we  can  combine  alternative  equalities,  we  can 
also  combine  simultaneous  inequalities,  since  they  are  redu- 
cible to  equalities. 

For,  from  the  formulas  previously  established  (S  57) 
{ab  =  o)  =  (a  =  6)  -\-  {b  =  o), 
(a  +  b=  i)  =  (a=  i)  +  (b=  i), 
we  deduce  by  contraposition 

(a  +  o)  {b^o)  =  (ab^o), 
(a+i)  (b+i)  =  {a  +  b+i). 

These  two  formulas,  moreover,  according  to  what  we  have 
just  said,  are  equivalent  to  the  known  formulas 

(a  =  i)  (b=i)^  (ab  =  i), 
(a  =  o)  (b  =="  o)  =  (a  +  b  =  o). 

Therefore,  in  the  calculus  of  propositions,  we  can  solve 
all  simultaneous  systems  of  equalities  or  inequalities  and  all 
alternative  systems  of  equalities  or  inequalities,  which  is  not 
possible  in  the  calculus  of  classes.  To  this  end,  it  is  necessary 
only  to  apply  the  following  rule: 

First  reduce  the  inclusions  to  equalities  and  the  non- 
inclusions  to  inequalities;  then  reduce  the  equalities  so  that  their 
second  members  will  be  i,  and  the  inequalities  so  that  their 
.second  members  will  be  o,  and  transform  the  latter  into  equal- 
ities having  i  for  a  second  member;  finally,  suppress  the 
second  members  i  and  the  signs  of  equality,  i.  <?.,  form  the 
product  of  the  first  members  of  the  simultaneous  equalities  and 
the  sum  of  the  first  members  of  the  alternative  equalities, 
retaining  the  parentheses. 


92  CONCLUSION. 

60.  Conclusion. — The  foregoing  exposition  is  far  from 
being  exhaustive;  it  does  not  pretend  to  be  a  complete 
treatise  on  the  algebra  of  logic,  but  only  undertakes  to  make 
known  the  elementary  principles  and  theories  of  that  science. 
The  algebra  of  logic  is  an  algorithm  with  laws  peculiar  to 
itself.  In  some  phases  it  is  very  analogous  to  ordinary  al- 
gebra, and  in  others  it  is  very  widely  different.  For  in- 
stance, it  does  not  recognize  the  distinction  of  degrees;  the 
laws  of  tautology  and  absorption  introduce  into  it  great 
simplifications  by  excluding  from  it  numerical  coefficients. 
It  is  a  formal  calculus  which  can  give  rise  to  all  sorts  of 
theories  and  problems,  and  is  susceptible  of  an  almost  in- 
finite development. 

But  at  the  same  time  it  is  a  restricted  system,  and  it  is 
important  to  bear  in  mind  that  it  is  far  from  embracing  all 
of  logic.  Properly  speaking,  it  is  only  the  algebra  of 
classical  logic.  Like  this  logic,  it  remains  confined  to  the 
domain  circumscribed  by  Aristotle,  namely,  the  domain  of 
the  relations  of  inclusion  between  concepts  and  the  relations 
of  implication  between  propositions.  It  is  true  that  classical 
logic  (even  when  shorn  of  its  errors  and  superfluities)  was 
much  more  narrow  than  the  algebra  of  logic.  It  is  almost 
entirely  contained  within  the  bounds  of  the  theory  of  the 
syllogism  whose  limits  to-day  appear  very  restricted  and 
artificial.  Nevertheless,  the  algebra  of  logic  simply  treats, 
with  much  more  breadth  and  universality^  problems  of  the 
same  order;  it  is  at  bottom  nothing  else  than  the  theory 
of  classes  or  aggregates  considered  in  their  relations  of  in- 
clusion or  identity.  Now  logic  ought  to  study  many  other 
kinds  of  concepts  than  generic  concepts  (concepts  of  classes) 
and  many  other  relations  than  the  relation  of  inclusion  (of 
subsumption)  between  such  concepts.  It  ought,  in  short,  to 
develop  into  a  logic  of  relations,  which  Leibniz  foresaw, 
which  Peirce  and  Schroder  founded,  and  which  Peano  and 
Russell  seem  to  have  established  on  definite  foundations. 

While  classical  logic  and  the  algebra  of  logic  are  of 
hardly  any  use  to  mathematics,  mathematics,  on  the  other 
hand,   finds  in   the  logic   of  relations  its   concepts   and   fun- 


CONCLUSION.  93 

damental  principles;  the  true  logic  of  mathematics  is  the  logic 
of  relations.  The  algebra  of  logic  itself  arises  out  of  pure 
logic  considered  as  a  particular  mathematical  theory,  for  it 
rests  on  principles  which  have  been  implicitly  postulated  and 
which  are  not  susceptible  of  algebraic  or  symbolic  expression 
because  they  are  the  foundation  of  all  symbolism  and  of  all 
the  logical  calculus.^  Accordingly,  we  can  say  that  the  al- 
gebra of  logic  is  a  mathematical  logic  by  its  form  and  by 
its  method,  but  it  must  not  be  mistaken  for  the  logic  of 
mathematics. 

I  The  principle  of  deduction  and  the  principle  of  substitution.  See 
the  author's  Manuel  de  Logistique,  Chapter  I,  §§  2  and  3  [not  published], 
and  Les  Principes  des  Maikimatiques,  Chapter  I,  A. 


INDEX. 


Absorption,  Law  of,   13,  92. 

Absurdity,  Type  of,  27. 

Addition,  and  multiplication, 
Logical,  V,  vi,  9,  20;  and 
multiplication,  Modulus  of, 
1 9 ;  and  multiplication,  The- 
orems on,  14;  Logical,  not 
disjunctive,  11. 

Affirmative  propositions,  80  n. 

Algebra,  of  logic  an  algorithm, 
92;  of  logic  compared  to 
mathematical  algebra,  13; 
of  thought,  V. 

Algorithm,  Algebra  of  logic 
an,  92. 

Alphabet  of  human  thought,  v. 

Alternative,  12;  affirmation, 
II,  20,  24;  Equivalence  of 
an  implication  and  an,  85. 

Antecedent,  7. 

Aristotle,  iii,  2 in.,  92. 

Assertion,  Principle  of,  84. 

Assertions,  Number  of  pos- 
sible, 79. 

Axioms,    8,    10,    16,    17,    22, 

27,  84. 

Baldwin,  23  n. 

Boole,  iii — ix,  xiii,  iin.,  18,  21, 

28,  29,    63;    Problem    of, 
59—61. 

Bryan,  William  Jennings,  ix. 


Calculus,  Infinitesimal,  v ;  Logic- 
al, viii,  3;  ratiocinator, 
V— viii. 

Cantor,  Georg,   ion. 

Categorical  syllogism,  8. 

Cause,  7,   II. 

Causes,  Forms  of,  69;  Law 
of,  67—69;  Sixteen,  67,  72; 
Table  of,  77—79. 

Characters,  v. 

Classes,  Calculus  of,  4,  86,  91. 

Classification  of  dichotomy, 
3  in. 

Commutativity,  24. 

Composition,  Principle  of, 
II— 12,  86. 

Concepts,  Calculus  of,  4. 

Condition,  7;  Necessary  and 
sufficient,  7-8,  45,  49,  57, 
82;  Necessary  but  not  suffi- 
cient, 41;  of  impossibility 
and  indetermination,  57, 

Connaissances,  63. 

Consequence,  7,  11. 

Consequences,  Forms  of,  69; 
Law  of,  63  —  66;  of  the 
syllogism,  9;  Sixteen,  65, 
71;  Table  of,  76-77. 

Consequent,  7. 

Constituents,  28;  Properties 
of,  29. 


INDEX. 


95 


Contradiction,     Principle    of,  I 

22n.,  23—24. 
Contradictory  propositions,  24 ;  ! 

terms,   29.  I 

Contraposition,    Law    of,    26,  ! 

81;  Principle  of,  88. 
Council,  Members  of,  71. 
Couturat,  v.,   i8n.,  93. 

Dedekind,   ion. 

Deduction,  61 ;  Principle  of,  93. 

Definition,  Theory  of,  x. 

De  Morgan,  iii,  iv,  vi,  viii,  ix; 
Formulas  of,  32—33,  81. 

Descartes,  iv. 

Development,  28;  Law  of, 
30 — 32  ;  of  logical  functions, 
79;  of  mathematics,  iv;  of 
symbolic  logic,  viii. 

Diagrams  of  Venn,  Geometric- 
al, 73-74- 

Dichotomy,  Classification  of, 
3  in. 

Disjunctive,  Logical  addition 
not,   11;  sums,  34. 

Distributive  law,   16. 

Double  inclusion,  37;  ex- 
pressed by  an  indeterminate, 
48;  Negative  of  the,   82. 

Double  negation,   24. 

Duality,  Law  of,  20. 

Economy  of  mental  eflfort,  iii. 

Elimination  of  known  terms, 
63,  64 — 67 ;  of  middle  terms, 
61,  63;  of  unknowns,  53, 
57,  59,  61;  Resultant  of, 
40,  41,57,  72,  73,82;  Rule 
for  resultant  of,  43,  55. 


Equalities,  Formulas  for  trans- 
forming inclusions  into,  15, 
25—26;  Reduction  of  in- 
equalities to,  85,  91. 

Equality  a  primitive  idea,  15; 
Definition  of,  6—8;  Notion 
of,  ix. 

Equation,  and  an  inequation, 
83;  Throwing  into  an,  75. 

Equations,  Solution  of,  50—53, 
57—59,  61,  73. 

Excluded  middle.  Principle  of, 
23—24. 

Exclusion,  Principle  of,  23  n. 

Exclusive,  Mutually,  29. 

Existence,  Postulate  of,  21,  27. 

Exhaustion,  Principle  of,  23  n. 

Exhaustive,  Collectively,  29. 

Forms,  Law  of,  62,  70;  of  con- 
sequences and  causes,  69. 

Frege,  vii,  viii,  x;  Symbolism 
of,  vii. 

Functions,  iv;  Development  of 
logical,  79;  Integral,  29 n; 
Limits  of,  37—38;  Logical, 
29—30;  of  variables,  56; 
Properties  of  developed, 
34—37;  Propositional,  iv; 
Sums  and  products  of,  44; 
Values  of,   55. 

Hopital,  Marquis  de  1',  vi. 
Huntington,  E.  V.,    xiv,    4n., 

i5n.,  2 in. 
Hypothesis,   7. 
Hypothetical   arguments,    27; 

reasoning,  89;  syllogism,  8. 


96 


INDEX. 


Ideas,  Simple  and  complex,  v. 

Identity,  vi;  Principle  of,  8, 
21,  88;  Type  of,  27. 

Ideography,  v,  vii,  viii. 

Implication,  5;  and  an  alter- 
native. Equivalence  of  an, 
85;  Relations  of,  92. 

Importation  and  exportation, 
Law  of,  88. 

Impossibility,  Condition  of,  57. 

Inclusion,  vi;  a  primitive  idea, 
ix,  5;  Double,  37;  expressed 
by  an  indeterminate,  46,  48; 
Negative  of  the  double,  82; 
Relation  of,  x,  4—6,  92. 

Inclusions  into  equalities.  For- 
mulas for  transforming,  15, 
25—26. 

Indeterminate,  5 1 ;  Inclusion 
expressed  by  an,  46,  48. 

Indetermination,  43;  Condition 

of;  57- 

Inequalities,  to  equalities.  Re- 
duction of,  85,  91;  Trans- 
formation of  non-inclusions 
and,  81. 

Inequation,  Equation  and  an, 
83;   Solution  of  an,  81,  84. 

Infinitesimal  calculus,  v. 

Integral  function,  29n. 

Interpretations  of  the  cal- 
culus, 3f. 

Jevons,  viii,  ix,  xiii,  iin.,  73; 

Logical  piano  of,  75. 
Johnson,  W.  E,,  xiv,   59. 

Known  terms  (connaissances)^ 
63-64,  67. 


Ladd-Franklin,  Mrs.,  viii,  xiii, 

23  n.,  42. 
Lambert,  iii,  vi. 
Leibniz,  iii,  ivff.,  4,  92. 
Limits  of  a  function,  37—38. 

MacCoU,  vi,  ix,   21  n.,  30. 

MacFarlane,  Alexander,  xiii. 

Mathematical  function,  ix; 
logic,  iii,  iv,  93. 

Mathematics,  Philosophy  a 
universal,  iv. 

Maxima  of  discourse,  29. 

Middle,  Principle  of  excluded, 
23—24;  terms,  Elimination 
of,  61,  63. 

Minima  of  discourse,   28. 

Mitchell,  O.,  xiii,  42. 

Modulus  of  addition  and  mul- 
tiplication,  19. 

Modus  ponens,  89. 

Modus  to  liens,   27,  89. 

Muller,  Eugen,  ix,  xiv,  46  n. 

Multiplication.  See  s.  v.  "Ad- 
dition." 

Negation,    v,   vi,   9;    defined, 
21-23;  Double,  24;  Dual- 
ity not  derived  from,  20,  22. 
Negative,  21,  23;  of  the  double 
inclusion,  82;  propositions. 
Son. 
Non-inclusions    and    inequal- 
I       ities,  Transformation  of,  81. 
I  Notation,  v,  2 in,  44. 
:  Null-class,  vi,  18,  20. 
Number  of  possible  assertions, 

'       79- 


INDEX. 


97 


One,  Definition  of,  ix,  17—20. 

Particular  propositions,  80. 

Peano,  iii,  viii,  x,  Son,  92. 

Peirce,  C.  S.,  viii,  ix,  xiii. 

Philosophy  a  universal  math- 
ematics, iv. 

Piano  of  Jevons,  Logical,  75. 

Poretsky,  xiv,  28,  73,  52n, 
53;  Formula  of,  38-39, 
40;  Method  of,  62  —  70. 

Predicate,  7. 

Premise,   7. 

Primary  proposition,  6,  21. 

Primitive  idea,  Equality  a,  15; 
Inclusion  a,  ix,  5. 

Product,  Logical,  10. 

Propositions,  ix;  Calculus  of, 
4,  86,  91;  Contradictory, 
24;  Formulas  peculiar  to 
the  calculus  of,  84;  Implica- 
tion between,  92;  reduced 
to  lower  orders,  2>^;  Un- 
iversal and  particular,  44,80. 

Reciprocal,   7,   21. 
Reduciio  ad  absurdum,  27. 
Reduction,  Principle  of,  89 n. 
Relations,  Logic  of,  92. 
Relatives,  Logic  of,  ix. 
Resultant    of   elimination,  40, 

41,   57,   72,   73;  82;    Rule 

for,  43,  55. 
Russell,  B.J  vii,  viii,  89n,  92. 

Schroder,  vi,  viiif,  xiii,  5,  2 in, 
29,  41.  59»  61—62,  8on, 
92;  Theorem  of,  39. 

Secondary  proposition,  6,  21. 


\  Simplification,     Principle     of, 

II  — 12,  21. 
i  Simultaneous   affirmation,    11, 
20,  24. 

Solution  of  equations,  50—53, 
57—59,  61,  73;  of  in- 
equations, 81,  84. 

Subject,  7. 

Substitution,  Principle  of,  93. 

Subsumption,  5. 

Summand,  4. 

Sums,  and  products  of  func- 
tions, 44;  Disjunctive,  34; 
Logical,   I  o. 

Syllogism,  Principle  of  the,  8, 
15,  62n;  Theory  of  the,  92. 

Symbolic  logic,  iii,  v;  Devel- 
opment of,  viii. 

Symbolism  in  mathematics,  iv. 

Symbols,  Origin  of,  iv. 

Symmetry,  7,  20,  24. 

Tautology,  Law  of,   13,  92. 

Term,  4. 

Theorem,  7. 

Thesis,  7. 

Thought,  Algebra  of,  v;  Alpha- 
bet of  human,  v;  Economy 
of,  iii. 

Transformation  of  inclusions 
into  equalities,  15,  25—26; 
of  inequalities  into  equal- 
ities, 85,  91;  of  non-in- 
clusions and  inequalities,  81. 

Universal  characteristic  of 
Leibniz,  V— viii;  propositions. 
Son. 


98 


INDEX. 


Universe    of    discourse,     i8, 

23n,  27. 
Unknowns,  Elimination  of,  53, 

57,  59,  61. 

Variables,  Functions  of,  56. 

Venn,  John,  iii,  viii,  ix;  Geo- 
metrical diagrams  of,  7  3  -  7  4 ; 
Mechanical  device  of,  75; 
Problem  of,  71—73. 


Viete,   iv, 
Voigt,  42. 

Whitehead,    A.    N.,    viii,    xiii, 

56n.,   59,  61  n. 
Whole,  Logical,  62. 

Zero,  Definition  of,  ix,  17  —  20; 
Logical,  62,  76. 


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